teorija aproksimacija za studente v semestar

TEORIJA APROKSIMACIJAza studente V semestraPMF u NisuPreuzeto iz rukopisa:

Interpolation Processes: BasicTheory and Applications

G. Mastroianni – G.V. Milovanovic

December 5, 2004

Springer-VerlagBerlin Heidelberg NewYorkLondon Paris TokyoHong Kong BarcelonaBudapest

Contents

1. Constructive Elements and Approaches in Approximation Theory. . 11.1 INTRODUCTION TOAPPROXIMATION THEORY . . . . . . . . . . . . . . . . 1

1.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Algebraic and trigonometric polynomials . . . . . . . . . . . . . . . . 31.1.3 Best approximation by polynomials . . . . . . . . . . . . . . . . . . . . . 71.1.4 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Chebyshev extremal problems . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.6 Chebyshev alternation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 161.1.7 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 CHEBYSHEV SYSTEMS AND INTERPOLATION . . . . . . . . . . . . . . . . . 221.2.1 Chebyshev systems and spaces . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.2 Algebraic Lagrange interpolation . . . . . . . . . . . . . . . . . . . . . . . 23

1.3 INTERPOLATION BY ALGEBRAIC POLYNOMIALS . . . . . . . . . . . . . . 241.3.1 Representations and computation of interpolation polyno-

mials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.2 Interpolation array and Lagrange operators . . . . . . . . . . . . . . . 261.3.3 Interpolation error for some classes of functions . . . . . . . . . . 291.3.4 Uniform convergence in the class of analytic functions . . . . . 321.3.5 Bernstein’s example of pointwise divergence . . . . . . . . . . . . . 361.3.6 Lebesgue function and some estimates for the Lebesgue

constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2. Orthogonal Polynomials and Weighted Polynomial Approximation. . 392.1 ORTHOGONAL SYSTEMS AND POLYNOMIALS . . . . . . . . . . . . . . . . . 39

2.1.1 Inner product space and orthogonal systems . . . . . . . . . . . . . . 392.1.2 Fourier expansion and best approximation . . . . . . . . . . . . . . . 412.1.3 Examples of orthogonal systems . . . . . . . . . . . . . . . . . . . . . . . 432.1.4 Basic facts on orthogonal polynomials and extremal problems 442.1.5 Zeros of orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 48

2.2 ORTHOGONAL POLYNOMIALS ON THE REAL L INE . . . . . . . . . . . . . 492.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3 CLASSICAL ORTHOGONAL POLYNOMIALS . . . . . . . . . . . . . . . . . . . . 552.3.1 Definition of the classical orthogonal polynomials . . . . . . . . . 552.3.2 General properties of the classical orthogonal polynomials . . 58

VI Contents

2.3.3 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1. Constructive Elements and Approachesin Approximation Theory

1.1 INTRODUCTION TOAPPROXIMATION THEORY

1.1.1 Basic notions

One of the main problems in approximation theory is how to find, for a given func-tion f from a large spaceX, a simple functionϕ from some small subsetΦ of X,such thatϕ be close in certain sense tof . We say thatϕ is anapproximationor anapproximantto a given functionf . Usually,X is a normed linear space of functionsdefined on a given setA. For example,A can be a compact interval[a,b], the circleT, etc. We use the circleT in the periodic cases, when it represents the real lineRwith the identification of the points modulo2π. The normed space can be the spaceof continuous functionsC(A), m-times continuous-differentiable functionsCm(A),the spaceLp(A), and other Banach spaces. The spaceLp(A) is defined in the usualway

Lp(A) ={

f∣∣∣ ‖ f‖p :=

(∫

A| f (t)|pdt

)1/p

< +∞}

, 1≤ p < +∞.

If f is defined everywhere onA and ‖ f‖∞ := supt∈A

| f (t)| < +∞ we write f ∈L∞(A).

WhenA = T≡ [0,2π] we simply writeLp = Lp(T) andL∞ = L∞(T).The distance betweenf andϕ can be measured by the norm‖ f −ϕ‖. Then, a

distance betweenf ∈ X andΦ is determined by

E( f ) = infϕ∈Φ

‖ f −ϕ‖.

This infimumE( f ) is called thebest approximationof the function f by elementsfrom Φ in a given norm. If there exists an elementϕ∗ ∈Φ such that

E( f ) = minϕ∈Φ

‖ f −ϕ‖= ‖ f −ϕ∗‖, (1.1.1)

we say thatϕ∗ is anelement of the best approximation. The questions on existenceand uniqueness of such an element are essential. Also, algorithms for its finding areof a great importance from the numerical point of view.

2 1. CONSTRUCTIVEELEMENTS AND APPROACHES

If Φ is a finite dimensional subspace ofX, then for eachf ∈ X always exists anelement of the best approximation. Unfortunately the element of the best approxi-mation is not always unique. However, whenX is strictly normed space1, then thiselement is unique. Some important special cases will be considered in Section 1.4.The proofs of general results can be found for example in [279], [175], [71].

Very often we approximate the functionf by algebraic polynomialson A =[a,b], i.e. we takeΦ = Pn, wherePn is the set of all algebraic polynomialspn ofdegree at mostn,

pn(x) =n

∑k=0

akxk (ak ∈ R). (1.1.2)

If the coefficientsak ∈ C, the corresponding set will be denoted byPCn . If an 6= 0,the degree ofpn is strictly n. A polynomial ismonicif an = 1.

For the circleT, i.e., in a periodic case, we takeΦ = Tn, whereTn is the set ofall trigonometric polynomialstn of degree at mostn,

tn(x) =a0

2+

n

∑k=1

(ak coskx+bk sinkx

)(ak,bk ∈ R). (1.1.3)

If |an|+ |bn|> 0 the degree oftn is strictly n.There are two important particular cases of (1.1.3). Namely, ifb1 = · · ·= bn = 0

we have acosinepolynomial

cn(x) = a0 +a1cosx+ · · ·+ancosnx.

Fora0 = a1 = · · ·= an = 0 it is asinepolynomial

sn(x) = b1sinx+ · · ·+bnsinnx.

Defining the complex coefficientsck (|k| ≤ n) as

c0 =a0

2, ck = c−k =

12(ak− ibk) (k = 1, . . . ,n),

we obtain the complex form of (1.1.3),

tn(x) =n

∑k=−n

ckeikx. (1.1.4)

Sometimes, we omitn in pn(x) andtn(x) and write simplep(x) andt(x) (or P(x)andT(x)), respectively.

Evidently,Pn is a vector space of dimensionn+1 overR. Therefore, this spaceequipped with any norm is isomorphic to the Euclidean vector spaceRn+1, and thesenorms are equivalent to each other.

Some usual norms of (1.1.2) on the spacePn areuniform or supremum normandLp-norm(p≥ 1), i.e.,

1 If ‖ f +g‖= ‖ f‖+‖g‖ implies thatf = αg (α ∈ R).

1.1 INTRODUCTION TOAPPROXIMATION THEORY 3

‖pn‖A = supx∈A

|pn(x)| and ‖pn‖p =(∫

A|pn(x)|pdx

)1/p

.

Similarly,Tn is a vector space of dimension2n+1. The previous norms are alsousual for trigonometric polynomials (1.1.3).

The main constructive elements in approximation theory are algebraic andtrigonometric polynomials and splines. In this book we deal only with polynomi-als. Two properties of polynomials are essential in approximation theory:

1◦ Each real continuous function on a finite closed interval can be uniformlyapproximated by polynomials.

2◦ Each polynomialpn ∈ Pn (tn ∈ Tn) can be uniquely interpolated atn+ 1(2n+1) points.

Regarding to the first property there are two basic theorems of Weierstrass.

Theorem 1.1.1.For each f ∈C[a,b] and eachε > 0 there is an algebraic polyno-mial p such that

| f (x)− p(x)|< ε (a≤ x≤ b).

Theorem 1.1.2.For each functionf ∈C(T) and eachε > 0 there is a trigonometricpolynomialt such that

| f (x)− t(x)|< ε (x∈ T).

Theorem 1.1.1 was first proved in 1885 by Weierstrass (see [353]). There areseveral different proofs of these theorems and their extensions and ramifications(see Lubinsky [195] and Pinkus [280]). Theorem 1.1.1 can be interpreted in termsof the best approximation in the uniform (supremum) norm‖ f‖[a,b] = max

a≤x≤b| f (x)|

( f ∈C[a,b]). LetEn( f ) = inf

p∈Pn‖ f − p‖[a,b]. (1.1.5)

Then, Theorem 1.1.1 asserts that

limn→+∞

En( f ) = 0, f ∈C[a,b].

The property2◦ of polynomials mentioned before enables another kind ofapproximation, which is known as theinterpolation of functions. This book ismainly devoted to the interpolation and interpolating processes in different func-tional spaces.

1.1.2 Algebraic and trigonometric polynomials

Consider an algebraic polynomialpn(z) of degreen,

pn(z) =n

∑k=0

αkzk (αk ∈ C).

The following result is well-known as thefundamental theorem of algebra(cf.[251, p. 177]):


Theorem 1.1.3.Every polynomial of degreen (≥ 1) with complex coefficients hasexactlyn zeros(counted with their multiplicities) in the complex plane.

The zeros of a polynomial are continuous functions of the coefficients of thepolynomial (see [251, pp. 177–178]).

Takingzon the circumference|z|= 1, i.e.,z= eiθ , pn(z) becomes a trigonomet-ric polynomialtn(θ) of degreen,

tn(θ) = pn(eiθ ) =a0

2+

n

∑k=1

(ak coskθ +bk sinkθ), (1.1.6)

with complex coefficients in the general case.The following important result is known as theHaar property.

Theorem 1.1.4.An arbitrary trigonometric polynomialtn(θ) of degree at mostn,which is not identically zero, cannot have more than2n distinct zeros inT (i.e., inany interval[a,a+2π), a∈ R).

Proof. Puttingz= eiθ and using Euler’s formulas

coskθ =12

(eikθ +e−ikθ ), sinkθ =12i

(eikθ −e−ikθ ),

we obtain

tn(θ) =n

∑k=−n

cn+keikθ ,

where

cn+k =

12(a−k + ib−k), k < 0,

12

a0, k = 0,

12(ak− ibk), k > 0.

Thus we haveeinθ tn(θ) = q(z), (1.1.7)

whereq is an algebraic polynomial of degree at most2n. If tn(θ) 6≡ 0, thentn(θ)cannot have more than2n distinct real zeros inT. ut

If the polynomialq(z) in (1.1.7) is of degree strictly2n, then it has exactly2n zeros in the complex plane (counted with their multiplicities). Denote them withz1, . . . ,z2n. In that casetn(θ) has also exactly2n zeros in any stripa≤Reθ < a+2π(a∈ R) of the complex plane.

Using a factorization ofq(z) and puttingzk = eiθk (k = 1, . . . ,2n), we get

tn(θ) = c2ne−inθ2n

∏k=1

(eiθ −eiθk

)

= c2nexp( i

2

2n

∑k=1

θk

) 2n

∏k=1

(ei(θ−θk)/2−e−i(θ−θk)/2

),


i.e.,

tn(θ) = A2n

∏k=1

sinθ −θk

2, (1.1.8)

where

A = (−1)n22nc2nexp( i

2

2n

∑k=1

θk

).

Thus, eachtn ∈ Tn\Tn−1 can be represented in the form (1.1.8).Consider now only real trigonometric polynomials2. If we put

c0 =12

a0, ck = ak− ibk (k = 1, . . . ,n),

whereak,bk ∈ R, then we can represent a real trigonometric polynomial as the realpart of an algebraic polynomial on the|z|= 1. Namely,

tn(θ) = Re

{ n

∑k=0

ckzk}

z=eiθ= Re

{12

a0 +n

∑k=1

(ak− ibk)eikθ}

=12

a0 +n

∑k=1

(ak coskθ +bk sinkθ

).

On the other hand, since

Re

{ n

∑k=0

ckzk}

=12

( n

∑k=0

ckzk +

n

∑k=0

ckzk)

we have

tn(θ) =12

z−nn

∑k=0

(ckz

n+k + ckzn−k

) ∣∣∣z=eiθ

,

i.e.,

tn(θ) =12

e−inθ q(eiθ ),

where

q(z) =n

∑k=0

(ckz

n+k + ckzn−k

)

= cn + · · ·+ c1zn−1 +2c0zn +c1zn+1 + · · ·+cnz2n.

Notice thatq(z) = z2nq(1/z), i.e., thatq is a self-inversive polynomial of degree2n(cf. [251, pp. 16–18]). According to the above we conclude that

|tn(θ)|= 12|q(eiθ )|.

2 Polynomials with real coefficients.


Two simple, but important real trigonometric polynomials are

cosnθ andsin(n+1)θ

sinθ.

Both of them can be expressed incosθ as algebraic polynomials of degreen. Puttingx = cosθ we obtain the well-known Chebyshev polynomials of the first and secondkind,

Tn(x) = Tn(cosθ) = cosnθ and Un(x) = Un(cosθ) =sin(n+1)θ

sinθ,

respectively. Their algebraic representations for|x| ≤ 1 are

Tn(x) = cos(narccosx) and Un(x) =sin

((n+1)arccosx

)√

1−x2.

Remark 1.1.1.For Chebyshev polynomials the following sums

12

+n

∑k=1

T2k(x) =12

U2n(x) andn

∑k=1

T2k−1(x) =12

U2n−1(x) (1.1.9)

hold.

Remark 1.1.2.If we puty = sin(θ/2), then

T2n+1(y)y

=cos[(2n+1)arccosy]

y=

cos[(2n+1)(π/2−arcsiny)]y

= (−1)n sin[(2n+1)arcsiny]y

,

i.e.,

T2n+1(y)y

= (−1)nsin

(2n+1)θ2

sinθ2

= (−1)nU2n

(cos

θ2

). (1.1.10)

According to (1.1.9) we get

T2n+1(y)y

= (−1)n

(1+2

n

∑k=1

coskθ

), y = sin

θ2

, (1.1.11)

becauseT2k(cos(θ/2)

)= coskθ . Thus, (1.1.10) is an even trigonometric polyno-

mial of degreen.

The Chebyshev polynomials will be treated in details later.


1.1.3 Best approximation by polynomials

In Subsection 1.1.1 we defined best approximation of the functionf ∈ X by ele-ments from some subsetΦ of X in a given norm. Here we deal with normed spacesC[a,b] andLp[a,b] (p≥ 1) and with their subsetsPn andTn (sets of algebraic andtrigonometric polynomials of degree at mostn, respectively).

Let Φ = Pn. According to the fact thatPn is a finite dimensional subspace ofX(X = C[a,b] or X = Lp[a,b], 1≤ p < +∞), the following result holds:

Theorem 1.1.5.Let f ∈ X, whereX = C[a,b] or X = Lp[a,b], 1≤ p < +∞. Thenfor eachn∈N there exists an algebraic polynomialP∗ (∈Pn) of best approximationin Pn for the functionf .

Usually we say that such a polynomial isbest uniform approximation(X =C[a,b]) or bestLp-approximation(X = Lp[a,b]).

A similar situation is appeared in the periodic case, when we takeΦ = Tn.

Theorem 1.1.6.Let f ∈ X, whereX = C[0,2π] or X = Lp[0,2π], 1 ≤ p < +∞.Then for eachn ∈ N there exists a trigonometric polynomialT∗ (∈ Tn) of bestapproximation inTn for the functionf .

Since the spacesLp[a,b] (in periodic caseLp[0,2π]) for 1< p< +∞ are strictlynormed (cf. [273]), then the polynomials of bestLp-approximations in such casesare unique. On the other side, the spacesL1[a,b] andC[a,b] are not strictly normed,so that inL1[a,b] we have not uniqueness of bestL1-approximation. As an illustra-tion of this fact is the following example:

Example 1.1.1.Let

f (x) =

{0, 0≤ x≤ 1,

1, 1 < x≤ 2.

Consider bestL1-approximation of this function in the set of all algebraic polyno-mials of degree zero. Since, for eachc (0≤ c≤ 1)

∫ 2

0| f (x)−c|dx=

∫ 1

0cdx+

∫ 2

1(1−c)dx= 1,

we conclude that every such constantc is bestL1-approximation inP0.

However, the situation is something different in the space of continuous func-tions on[a,b]. Namely, as a consequence of the well-knowChebyshev alternationtheorem(see Subsection 1.1.6), best uniform approximationP∗ (∈ Pn) for a func-tion f ∈C[a,b] is unique.

Instead of the termuniform approximation, we use alsoChebyshev approxi-mation. The concept of best approximation was introduced mainly by the work ofthe famous Russian mathematician Pafnutiı L’vovich Chebyshev (1821–1894), whostudied properties of polynomials with least deviation from a given continuous func-tion (see [49, 50]).


Let f ∈C[a,b] and‖ f‖= ‖ f‖∞ = ‖ f‖[a,b] = maxa≤x≤b

| f (x)|. As before, according

to (1.1.1), i.e., (1.1.5), we put

En( f ) = minp∈Pn

‖ f − p‖= ‖ f −P∗‖. (1.1.12)

In particular, for the functionf (x) = xn+1 on [−1,1], it is clear that (1.1.12)becomes

En(xn+1) = minp∈Pn

‖xn+1− p(x)‖= minq∈Pn+1

‖q‖= ‖Q∗n+1‖, (1.1.13)

wherePn+1 denotes the set of all monic polynomials of degreen+ 1 andQ∗n+1 is

the monic polynomial of least uniform norm on[−1,1] among all polynomials ofdegreen+ 1, with leading coefficient unity. In this way, with fixed leading coeffi-cient, Chebyshev [49] introduced polynomials of least deviation from zero, whichare known today as Chebyshev polynomials of the first kindTn(x). We mentionedsuch polynomials on the end of Section 1.1.2 in a connection with a simple trigono-metric polynomial. Precisely, Chebyshev showed thatQ∗

n(x) = 21−nTn(x). The no-tationcos(narccosx) = Tn(x) was introduced by Bernstein3.

The polynomialsTn(x) appear prominently in various extremal problems withpolynomials (cf. [251]). The following subsection deals with the Chebyshev poly-nomials.

1.1.4 Chebyshev polynomials

Basic properties. As we mentioned before, the Chebyshev polynomials of the firstand second kind can be expressed for|x| ≤ 1 in the forms

Tn(x) = cos(narccosx) and Un(x) =sin

((n+1)arccosx

)√

1−x2, (1.1.14)

respectively. It is easy to see thatT0(x) = 1, T1(x) = x andU0(x) = 1, U1(x) = 2x.Also, using

cos(n+1)θ +cos(n−1)θ = 2cosθ cosnθ ,

with x = cosθ , we see that the polynomialsTn satisfy the recurrence relation

Tn+1(x) = 2xTn(x)−Tn−1(x), n = 1,2, . . . . (1.1.15)

The same recurrence relation also holds for polynomials of the second kind,i.e., Un+1(x) = 2xUn(x)−Un−1(x), n≥ 1. Starting fromT0(x) = 1 andT1(x) = xor U0(x) = 1 and U1(x) = 2x, we compute the both sequences of polynomials{Tn(x)}+∞

n=0 and{Un(x)}+∞n=0 very easily. For example, forn = 0,1, . . . ,6 we get

3 It was derived from another transliteration of the name Chebyshev in the form Tchebychevor related forms.


T0(x) = 1, U0(x) = 1,T1(x) = x, U1(x) = 2x,T2(x) = 2x2−1, U2(x) = 4x2−1,T3(x) = 4x3−3x, U3(x) = 8x3−4x,T4(x) = 8x4−8x2 +1, U4(x) = 16x4−12x2 +1,T5(x) = 16x5−20x3 +5x, U5(x) = 32x5−32x3 +6x,T6(x) = 32x6−48x4 +18x2−1, U6(x) = 64x6−80x4 +24x2−1.

The relation (1.1.15) is known as thethree-term recurrence relation.

Sincex = cosθ and√

1−x2 = sinθ , using

cosnθ =12

(einθ +e−inθ

)=

12

[(cosθ + i sinθ)n +(cosθ − i sinθ)n] ,

we can get the following expressions for all complexx

Tn(x) =12

(ρn +ρ−n), Un(x) =

ρn+1−ρ−n−1

ρ−ρ−1 , (1.1.16)

where we put

ρ = x+√

x2−1 (x∈ C). (1.1.17)

The square root in (1.1.17) is such that|x+√

x2−1|> 1 wheneverx∈ C\ [−1,1].Forx∈ [−1,1] these formulas reduce to (1.1.14).

In the general case, the explicit expressions for Chebyshev polynomials of thefirst and second kind are

Tn(x) =n2

[n/2]

∑k=0

(−1)k(n−k−1)!k!(n−2k)!

(2x)n−2k (n≥ 1)

and

Un(x) =[n/2]

∑k=0

(−1)k(n−k)!k!(n−2k)!

(2x)n−2k ,

respectively.Chebyshev polynomialsTn(x) for n = 0,1, . . . ,6 are displayed in Fig. 1.1.1.SinceTn(x) = cos(narccosx) for −1≤ x≤ 1, it is easy to see that|Tn(x)| ≤ 1

for eachn≥ 0 and−1≤ x≤ 1 (see Fig. 1.1.1). Also, we have

|Un(x)| ≤ n+1 and |√

1−x2Un(x)| ≤ 1

for −1≤ x≤ 1. Some interesting values are:

Tn(±1) = (±1)n, T2n(0) = (−1)n, T2n+1(0) = 0,

T ′n(±1) = (±1)nn2, T(k)n (1) =

n2(n2−1) · · ·(n2− (k−1)2)(2k−1)!!

.


-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Fig. 1.1.1.The graphs ofy = Tn(x) for n = 0,1, . . . ,6 and−1≤ x≤ 1

Differential equation. Differentiating the relationy = Tn(x) we obtain thediffer-ential equation of the Chebyshev polynomials

(1−x2)y′′−xy′+n2y = 0.

The second particular solution of this equation,Sn(x) = sin(narccosx) (−1≤ x≤1) can be expressed in terms of the Chebyshev polynomials of the second kind.Namely,Sn(x) = Un−1(x)

√1−x2. The correspondingdifferential equation of the

Chebyshev polynomials of the second kindis

(1−x2)y′′−3xy′+n(n+2)y = 0.

Zeros and extremal points. According to (1.1.14) thezeros ofTn(x) can be ex-pressed in an explicit form,

xk = xn,k = cos(2k−1)π

2n(k = 1, . . . ,n). (1.1.18)

The zerosxk are real, distinct, and lie in(−1,1). In order to give a geometricinterpretation of zeros, we putθk = (2k−1)π/(2n), k= 1, . . . ,n. Now, it is clear thatthe zerosxk are the projections onto the real line of equally spaced pointsexp(iθk)on the upper arc of the unit circle. Thus, the zerosxk are more densely distributedaround endpoints±1 than in the interior of(−1,1). Precisely, using an idea from thetheory of probability, we can introduce thedistributionof zeros of the polynomialsTn(x) whenn tends to infinity, so-called thelimit distribution.

Let Nn(a,b) be the number of zeros ofTn(x) which are in[a,b) ⊂ [−1,1], i.e.,Nn(a,b) = {m∈ Zn |a≤ cosθk < b}, whereZn = {1, . . . ,n}. Then, the correspond-ing density functionat the pointx∈ (−1,1) is given by

ψ(x) = limh→0

1h

limn→+∞

Nn(x,x+h)n

.


It is not difficult to see thatNn(x,x+ h)− (n/π)(arccosx−arccos(x+ h)) is equalto 0 or−1, so that

limn→+∞

Nn(x,x+h)n

=arccosx−arccos(x+h)

π.

This means that we have so-calledarc sine distribution, because the density functionis given by

ψ(x) =1π

1√1−x2

(−1 < x < 1).

Thus, the limit distribution of zeros of the Chebyshev polynomials isdµ(x) =ψ(x)dx= π−1(1−x2)−1/2dx.

In order to provide the arrangement of the zeros,−1 < x1 < x2 < · · · < xn < 1,we change very oftenk by n−k+1 in (1.1.18) so that

xk = xn,k =−cos(2k−1)π

2n(k = 1, . . . ,n). (1.1.19)

Another interesting points are the points whereTn(x) =±1,

ξk = ξn,k =−coskπn

(k = 0,1, . . . ,n). (1.1.20)

The points (1.1.20) are known as theextremal points ofTn(x). Their limit distribu-tion is also the arc sine distribution on[−1,1].

Chebyshev polynomials on the complex plane.In order to investigateTn(z) for acomplexzoutside of the interval[−1,1], we need theJoukowski transformation

z=12

(w+

1w

), (1.1.21)

which maps|w|> 1 ontoC\ [−1,1] and maps the unit circle|w|= 1 onto the interval[−1,1]. Takingw = reiθ andz= x+ iy, we get

x+ iy =12

(reiθ +

1r

e−iθ)

=12

(r +

1r

)cosθ + i

12

(r− 1

r

)sinθ ,

i.e.,

x =12

(r +

1r

)cosθ , y =

12

(r− 1

r

)sinθ .

For a constantr > 1, these equations describe an ellipseEr : (x/a)2+(y/b)2 = 1,with the semi-axes

a =12

(r +

1r

), b =

12

(r− 1

r

)

and the foci±1 (a2−b2 = 1). For r = 1, Er reduces to the interval[−1,1].


Thus, for a givenz∈C\ [−1,1] there exists exactly one ellipseEr (r > 1) passingthroughz, wherer is determined by

r +1r

= 2a = |z+1|+ |z−1|.According to (1.1.16) and (1.1.21) we have

Tn(z) =12

[(z+

√z2−1

)n +(z−

√z2−1

)n], (1.1.22)

where

z=12

(w+w−1) =

12

(reiθ +

1r

e−iθ)

,

i.e.,z+√

z2−1= reiθ , z−√

z2−1= r−1e−iθ . Here,|z+√

z2−1|= r > 1 wheneverz∈ C\ [−1,1].

Using (1.1.22) we find

|Tn(z)|= 12

√r2n +2cos2nθ +

1r2n , (1.1.23)

as well as

ReTn(z) =12

(rn +

1rn

)cosnθ , ImTn(z) =

12

(rn− 1

rn

)sinnθ .

As we know, the all zeros ofTn are inside(−1,1), but it is interesting to mentionthat for a fixedz, arbitrarily close to the interval[−1,1], the sequence

{|Tn(z)|}

n∈Ntends to infinity with geometric rate. Indeed, from (1.1.23) we find

limn→+∞

|Tn(z)|1/n = r (z∈ C\ [−1,1]). (1.1.24)

Basing on the above, (1.1.24) holds for eachz∈ Er (r > 1).

Some other relations. It is easy to prove the following relations:

Un(x) =1

n+1T ′n+1(x) =

11−x2

(xTn+1(x)−Tn+2(x)

),

Tn(x) = Un(x)−xUn−1(x) = xUn−1(x)−Un−2(x),

Un(x) =1

2(n+1)(U ′

n+1(x)−U ′n−1(x)

),

U ′n(x) =

11−x2

((n+1)Un−1(x)−nxUn(x)

)

Also, for 2≤ k≤ n we can check

Tn(x) = Tk(x)Un−k(x)−Tk−1(x)Un−k−1(x).

Indeed, using (1.1.16), the right hand side in this equality reduces to

12

(Un(x)−Un−2(x)

)= xUn−1(x)−Un−2(x) = Tn(x).


Orthogonality. The sequences{Tn(x)}+∞n=0 and {Un(x)}+∞

n=0 are orthogonalon(−1,1) in the following sense

∫ 1

−1Tn(x)Tm(x)

dx√1−x2

= 0,∫ 1

−1Un(x)Um(x)

√1−x2dx= 0 (n 6= m).

They are special cases of the so-called Jacobi polynomials{P(α,β )n (x)}+∞

n=0, withparametersα,β > −1. The Chebyshev polynomials of the first and second kindcorrespond to parametersα = β =−1/2 andα = β = 1/2, respectively. The Jacobipolynomials and other orthogonal polynomials will be treated in Chapter 2.

1.1.5 Chebyshev extremal problems

We start this subsection by considering the following extremal problem:Among allpolynomials of degreen, with leading coefficient unity, find the polynomial whichdeviates least from zero in a given norm‖ .‖.The extremal problem in the uniform norm. As we mentioned before, Cheby-shev [49] solved the previous problem in the uniform norm‖ f‖[−1,1] = max

−1≤x≤1| f (x)|.

Let Tn(x) be the monic Chebyshev polynomial of the first kind of degreen, i.e.,

T0(x) = T0(x) = 1, Tn(x) =1

2n−1 Tn(x) (k = 1,2, . . .),

whereTn(x) = cos(narccosx) for −1≤ x≤ 1.

Theorem 1.1.7.Letq(x) =n∑

ν=0aνxν , withan = 1, be an arbitrary monic polynomial

of degreen. Then

‖q‖[−1,1] ≥ ‖Tn‖[−1,1] =

{21−n, n > 0,

1, n = 0,(1.1.25)

with equality only if q(x) = Q∗n(x) = Tn(x).

Proof. Settingn instead ofn+ 1 in (1.1.13) we see that forn = 0, the statement(1.1.25) is trivial. Therefore, we must consider the casen > 0.

Firstly, we note that the extremal points ofTn(x), given by (1.1.20), are ordered,i.e.,

−1 = ξ0 < ξ1 < · · ·< ξn = 1.

Let r(x) = Tn(x)− q(x) be a polynomial of degree at mostn− 1. In order toprove (1.1.25) we suppose on the contrary that

Q = ‖q‖[−1,1] < ‖Tn‖[−1,1] = 21−n. (1.1.26)

Let x∈ [−1,1]. According to (1.1.26) we have


−21−n−q(x) ≤ −21−n +Q < 0,

21−n−q(x) ≥ 21−n−Q > 0,

from which we conclude that the polynomialr(x) has alternatively positive andnegative values in the pointsξk (k = 0,1, . . . ,n), given by (1.1.20). Therefore, thispolynomial must have at leastn zeros, which is a contradiction, becauser ∈ Pn−1.

utFrom this important theorem we conclude (cf. Rivlin [294, 295]):(a) The polynomialP∗ ∈ Pn−1 closest to the power functionx 7→ f (x) = xn,

where closeness is measured by‖ f − p‖[−1,1] (p∈ Pn−1), is

P∗(x) = xn− Tn(x).

(b) LetP(x) =n∑

ν=0aνxν be an arbitrary algebraic polynomial of degreen and let

F :Pn → R be a linear functional defined by

F(P) = an =1n!

P(n)(0).

Among allP∈ Pn satisfying‖P‖[−1,1] = 1, the largest value of|F(P)| is 2n−1, andthis value is attained only forP(x) =±Tn(t).

This fact can be expressed as an inequality for the leading coefficient of thepolynomialP∈ Pn, i.e.,

|an| ≤ 2n−1‖P‖[−1,1]. (1.1.27)

This is the well-knownChebyshev inequality.

Remark 1.1.3.For polynomialsP∈ Pn, van der Corput and Visser [53] proved that

max−1≤t≤1

|P(t)|2− min−1≤t≤1

|P(t)|2 ≥ |an|222n−2 .

This inequality contains the Chebyshev inequality (1.1.27).

It is interesting to mention that the Chebyshev polynomialTn(x) has also anextremal property in the following sense:

Theorem 1.1.8. If P∈ Pn such that|P(x)| ≤ 1 for −1≤ x≤ 1, then

|P(x)| ≤ |Tn(x)| for |x| ≥ 1.

Namely, the Chebyshev polynomialTn(x) is the fastest growing polynomial out-side[−1,1] among all polynomials of degreen, with ‖P‖[−1,1] ≤ 1. More general,we have (see Rivlin [295, p. 93]):

Theorem 1.1.9. If P∈ Pn and max0≤k≤n

|P(ξk)| ≤ 1, whereξk are extremal points de-

fined by(1.1.20), then form= 0,1, . . . ,n,

|P(m)(x)| ≤ |T(m)n (x)| for |x| ≥ 1. (1.1.28)

Equality is possible in(1.1.28) for m≥ 1 and|x|> 1 only if P(x) =±Tn(x).

The other results in this direction can be found in [251, Chapter 5].


The extremal problem in L1-norm. An analogous result to (1.1.25) inL1-norm,

‖ f‖1 =∫ 1

−1| f (x)|dx, (1.1.29)

was proved by Korkin and Zolotarev [174]:

Theorem 1.1.10.Let q(x) =n∑

ν=0aνxν , with an = 1, be an arbitrary monic polyno-

mial of degreen. Then‖q‖1 ≥ ‖Un‖1 = 21−n,

with equality only ifq(x) = Un(x), whereUn is the monic Chebyshev polynomial ofthe second kind of degreen, i.e.,Un(x) = 2−nUn(x).

Proof. Using the norm (1.1.29), we define the functionalJ:Pn → R by

J(q) = ‖q‖1 =∫ 1

−1|q(x)|dx.

Since ∫ 1

−1xksgnUn(x)dx=

{0, 0≤ k≤ n−1,

21−n, k = n.

(see [251, pp. 408–409]), for the monic polynomialq(x), we have

∫ 1

−1q(x)sgnUn(x)dx=

12n−1 ,

from which it follows that

J(q) =∫ 1

−1|q(x)|dx≥ 1

2n−1 .

Also, for the monic polynomialUn(x) = 2−nUn(x), we have

∫ 1

−1|Un(x)|dx=

∫ 1

−1

12n Un(x)sgnUn(x)dx=

12n−1 .

Thus, the polynomialUn(x) minimizes the functionalJ(q) = ‖q‖1.It remains to prove that this polynomial is the only polynomial which minimizes

the functionalJ. For this let us suppose that there exists another monic polynomial,sayR(x), of degreen, such thatJ(R) = 21−n. Then, it follows thatR(x) ≡ Un(x),which is a contradiction. ut

Similar extremal problems can be considered also in another norms (cf. [251,Chapter 5]).


1.1.6 Chebyshev alternation theorem

Let f ∈C[a,b]. As before, we put‖ f‖= ‖ f‖∞ = ‖ f‖[a,b] = maxa≤x≤b

| f (x)| and

En( f ) = minp∈Pn

‖ f − p‖= ‖ f −P∗‖. (1.1.30)

The polynomialP∗(x) of best uniform approximation can be characterized bythealternation theorem, which was proved by Chebyshev [49] in the case whenfis a differentiable function. A complete proof of this important theorem was givenindependently by Blichfeldt [37] and Kirchberger [169] on the beginning of the 20thcentury.

Theorem 1.1.11.If f ∈ C[a,b], then P∗ ∈ Pn is the polynomial of best uniformapproximation tof if and only if there existn+2 pointsx0, x1, . . ., xn+1 (a≤ x0 <x1 < · · ·< xn+1 ≤ b) such that

f (xk)−P∗(xk) = ε(−1)k‖ f −P∗‖ (k = 0,1, . . . ,n+1), (1.1.31)

where the constantε is +1 or −1.

We say thatP∗ realizes theChebyshev alternationfor f in [a,b] if the condi-tions (1.1.31) are satisfied. The pointsxk in (1.1.31) are calledalternation pointsforthe approximating polynomial (approximant)P∗. Theorem 1.1.11, which is calledsometimes asequal ripple theorem, holds also for an arbitrary real Haar space (seeDefinition 1.2.1 in Subsection 1.2.1), taking it instead ofPn. The proof of this theo-rem (in original or generalized form) can be found in several books (cf. DeVore andLorentz [71], Feinerman and Newman [100], Korneichuk [175], Meinardus [229],Petrushev and Popov [279]).

As we mentioned before, the uniqueness of best uniform approximation is aconsequence of the Chebyshev alternation theorem.

Theorem 1.1.12.The polynomialP∗ ∈Pn of best uniform approximation to a givenfunction f ∈C[a,b] is unique.

Proof. Assume that for a functionf ∈ C[a,b] there are two polynomials of bestapproximation inPn, P∗ andP∗:

‖ f −P∗‖= ‖ f − P∗‖= En( f ).

Then alsoQ∗ = (P∗+ P∗)/2 (∈ Pn) is a polynomial of best uniform approximationfor f . It means that there existn+ 2 pointsxk in [a,b] (see Theorem 1.1.11) suchthat

f (xk)−Q∗(xk) =12

[(f (xk)−P∗(xk)

)+

(f (xk)− P∗(xk)

)]= ε(−1)kEn( f )

for k = 0,1, . . . ,n+1, whereε =±1. Since


| f (xk)−P∗(xk)| ≤ En( f ) and | f (xk)− P∗(xk)| ≤ En( f ),

the previous equality can be fulfilled only if

f (xk)−P∗(xk) = f (xk)− P∗(xk) (k = 0,1, . . . ,n+1),

i.e., if P∗(xk) = P∗(xk) for eachk = 0,1, . . . ,n+ 1. This means thatP∗ ≡ P∗, i.e.,the polynomial of best uniform approximation for a continuous function on[a,b] isunique. ut

Another useful application of the Chebyshev alternation theorem is the follow-ing lower estimate ofEn( f ), obtained by de la Vallee Poussin [338, p. 85].

Theorem 1.1.13.Let f ∈C[a,b], P∈ Pn, andn+2 pointsx0, x1, . . ., xn+1 be suchthat a≤ x0 < x1 < · · · < xn+1 ≤ b. If f (xk)−P(xk) = ε(−1)kµk, with ε = ±1 andµk > 0, k = 0,1, . . . ,n+1, then

En( f )≥ µ = min0≤k≤n+1

µk.

Proof. Let P∗ ∈ Pn be the polynomial of best uniform approximation tof and sup-pose that‖ f −P∗‖ = En( f ) < µ . Then, for the polynomialQ(x) = P(x)−P∗(x)(Q∈ Pn) we must have

sgnQ(xk) = sgn[(

f (xk)−P∗(xk))− (

f (xk)−P(xk))]

= ε(−1)k,

which means that the polynomialQ(x) has alternate signs atn+ 2 pointsxk, k =0,1, . . . ,n+1. Thus,Q(x) must have at leastn+1 different zeros in[a,b], which isa contradiction, becauseQ∈ Pn. Therefore,En( f )≥ µ . ut

This theorem is very useful in numerical methods for finding the polynomial ofbest uniform approximation.

Some classical special cases.We consider now two special cases when the polyno-mial of the best approximation and corresponding quantityEn( f ) can be determinedin an explicit form.

1. The first one is the well-known Chebyshev examplef (x) = xn+1 on [−1,1].Taking the extremal points ofTn+1(x) as the alternation pointsxk in (1.1.31), i.e.,

xk = ξ (n+1)k =−cos

kπn+1

(k = 0,1, . . . ,n+1), (1.1.32)

according to Theorems 1.1.11 and 1.1.12, we must havexn+1−P∗(x) = Tn+1(x),i.e.,

P∗(x) = xn+1− 12n Tn+1(x) and En(xn+1) =

12n .

Thus, we have just obtained the Chebyshev result given by Theorem 1.1.7.2. The second example is best uniform approximation of the function


f (x) =1

x−a(a > 1)

on the interval[−1,1] by polynomials inPn. It was first solved by Chebyshev.This problem was investigated also by Bernstein [31, pp. 120–121], who gave thetrigonometric representation

1x−a

−P∗(x) =(a−

√a2−1)n

a2−1cos(nϕ +δ ),

wherex = cosϕ and(ax−1)/(x−a) = cosδ . It is clear that

En

( 1x−a

)=

(a−√

a2−1)n

a2−1.

A more transparent solution was given by Achieser [6, pp. 69–71] in the follow-ing form:

1x−a

−P∗(x) =M2

{vn α−v

1−αv+v−n 1−αv

α−v

},

where

x =12

(v+

1v

), a =

12

(α +

1α

) (|v|= 1, |α|< 1)

and

M = En

( 1x−a

)=

4αn+2

(1−α2)2 =(a−

√a2−1)n

a2−1,

becauseα = a−√

a2−1. Some details of the solution can be found in [229, pp. 34–36].

1.1.7 Numerical methods

Several methods for numerically computing the best uniform polynomial approx-imation to a given continuous function on[a,b] are described in Meinardus [229,pp. 105–130]. In this subsection we mention only so-called Remez algorithms[288, 289], which can be applied also for generalized cases of arbitrary real Haarsubspaces (see Subsection 1.2.1), instead ofPn. Precisely, we will describe a variantof these algorithms which is known as thesecond Remez algorithm.

Without loss of generality, we consider the approximation problem for a contin-uous function on[−1,1]. For a such functionf ∈C[−1,1] and an algebraic polyno-

mial P∈ Pn, i.e.,P(x) =n∑

ν=0aνxν , we put

δn(x) = δn(x;a) = f (x)−n

∑ν=0

aνxν ,

wherea= (a0,a1, . . . ,an)∈Rn+1. According to the Chebyshev alternation theorem,we formulate the following algorithm:


1◦ Start withn+2 selected points{xk}n+1k=0, such that

−1≤ x0 < x1 < · · ·< xn+1 ≤ 1;

2◦ Find a0,a1, . . . ,an,E from the linear system of equations

δn(xk;a) = f (xk)−n

∑ν=0

aνxνk = (−1)kE (k = 0,1, . . . ,n+1); (1.1.33)

Because of (1.1.33) in each of intervals[xk,xk+1] the functionx 7→ δn(x) pos-sesses at least one zerozk (xk < zk < xk+1).

3◦ Determine the points{zk}n+1k=−1 such that

z−1 =−1, δn(zk) = 0 (k = 0,1, . . . ,n), zn+1 = 1;

4◦ Select the pointsxk ∈ [zk−1,zk], k = 0,1, . . . ,n+1, such that

(sgnδn(xk))δn(xk) = maxzk−1≤x≤zk

{sgnδn(xk)δn(x)

};

5◦ If ‖δn(·,a)‖ > max0≤k≤n+1

|δn(xk;a)| then there exists a pointx∈ [−1,1] such

that |δn(x;a)| = ‖δn(·,a)‖. In that case, put the pointx in place of some point in{xk}n+1

k=0 so that the functionx 7→ δn(x,a) would preserve the alternating signs on thesuch obtained new set of points{xk}n+1

k=0;

6◦ For a given toleranceε, check the condition|‖δn(·;a)‖− |E| | < ε. If thiscondition is satisfied then stop; otherwise, putxk:= xk (k = 0,1, . . . ,n+1) and goto 2◦.

Thus, as the best polynomial approximation to a given continuous function wetake the algebraic polynomial which satisfies the “tolerance condition” in the laststep of the algorithm. The algorithm is not essentially sensitive to the choice ofthe initial points{xk}n+1

k=0. Very often it is convenient to take the extremal points ofTn+1(x), i.e., (1.1.32).

A modified version of the second Remez algorithm for polynomial approxima-tion of differentiable functions was given by Murnaghan and Wrench [262] (see also[261]).

The linear convergence of the second Remez algorithm can be proved for eachcontinuous function (cf. [279, pp. 13–15]):

Theorem 1.1.14.Let f ∈C[−1,1] andP∗ ∈Pn be its polynomial of best uniformapproximation. The polynomialPν (∈Pn) generated on theν-th step by the secondRemez algorithm satisfies the condition

‖Pν −P∗‖ ≤Cρν ,

where0 < ρ < 1 andC is a constant independent ofν .


Under some restrictions on the smoothness of the functionf it is possible toprove the quadratic convergence of this algorithm (cf. Meinardus [229, pp. 111–113]).

In order to illustrate this Remez algorithm we give two examples.

Example 1.1.2.Consider a continuous function defined on[−1,1] by

f (x) :=√

3+2x+4x2 .

For this simple function we want to find its best polynomial approximation inthe set of all polynomials of degree at most three. Thus, in theν-th iteration we have

δ (ν)3 (x) = f (x)− (a(ν)

0 +a(ν)1 x+a(ν)

2 x2 +a(ν)3 x3).

Starting from the extremal points ofT4(x), i.e.,{−1,−√2/2,0,

√2/2,1

}, the

second Remez algorithm generates the sequences of polynomial coefficients, givenin Table 1.1.1. Also, the corresponding quantityE(ν) is presented in the last columnof this table.

Table 1.1.1.The polynomial coefficients and the quantityE(ν) in the second Remez algorithm

ν a(ν)0 a(ν)

1 a(ν)2 a(ν)

3 E(ν)

1 1.7510913738 0.5217341087 0.8859831812−0.1397680974 −0.0190405662

2 1.7494089696 0.5261703529 0.8893792012−0.1442043415 −0.0207541821

3 1.7494113495 0.5261479622 0.8893942621−0.1441819510 −0.0207716229

4 1.7494113492 0.5261479629 0.8893942627−0.1441819516 −0.0207716232

-1 -0.5 0 0.5 1

1.8

2

2.2

2.4

2.6

2.8

3

-1 -0.5 0 0.5 1

-0.02

-0.01

0

0.01

0.02

Fig. 1.1.2.The graphics ofx 7→ f (x) =√

3+2x+4x2 (solid line) andx 7→ P∗(x) (dashedline) for n = 3 (left) and the deviationδ ∗3 (x) = f (x)−P∗(x) (right)

Thus, as an approximate solution (rounded to 8 decimals) we can take

P∗(x) = 1.74941135+0.52614796x+0.88939426x2−0.14418195x3.


The alternation points are:x0 = 1, x1 = −0.72898482, x2 = −0.12747162, x3 =0.58607094, x4 = 1, and

E3( f ) = ‖ f −P∗‖ ≈ 2.077×10−2.

The corresponding graphics are displayed in Figure 1.1.2.

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

Fig. 1.1.3.The graphics of functionsx 7→ f (x) =√|sinπx/2| and x 7→ P∗(x) for n = 6

(dashed line) andn = 12 (solid line)

-1 -0.5 0 0.5 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-1 -0.5 0 0.5 1

-0.1

-0.05

0

0.05

0.1

Fig. 1.1.4.The deviationδ ∗n (x) = f (x)−P∗(x) for n = 6 (left) andn = 12 (right)

Example 1.1.3.Consider now a function defined byf (x) :=√|sin(πx/2)| on

[−1,1]. For its best polynomial approximation in the setP6 we get

P∗(x) = 0.17787718+5.93869742x2−12.34268353x4 +7.40398610x6

andE6( f ) = ‖ f −P∗‖ ≈ 0.1779. Similarly, in the setP12 we obtain

P∗(x) = 0.126045547094028+16.9552165863717x2−149.188517382840x4

+ 573.700153068282x6−1048.54708556994x8

+ 901.727265823760x10−293.899123618993x12,


with E12( f ) ≈ 0.1260. The corresponding graphics are presented in Figures 1.1.3and 1.1.4.

Remark 1.1.4.Some details on thefirst Remez algorithmcan be found in [279,pp. 10–12]. Several modifications of this algorithm for solving linear and nonlinearChebyshev approximation problems on compactB⊂ Rs, as well as their conver-gence, were studied by Reemtsen [287].

1.2 CHEBYSHEV SYSTEMS AND INTERPOLATION

1.2.1 Chebyshev systems and spaces

We start with the setPn of all algebraic polynomials of degree at mostn defined onA = [a,b]. In Section 1.1.1 we mentioned two essential properties of polynomialsin approximation theory. One of them is thateach algebraic polynomial of degreeat mostn−1 can be uniquely interpolated atn points(Note thatn is replaced byn−1).

This property is equivalent to the fact that anyp∈ Pn−1 that vanishes atn pointsis identically zero. This property can be translated to the other finite-dimensionalsubspaces ofC(A), whereA is a Hausdorff topological space.

Definition 1.2.1. Let ϕk:A→R (k= 1, . . . ,n) be continuous functions. The setH ={ϕ1, . . . ,ϕn} is called aChebyshev systemor Haar systemof dimensionn on Aif Xn = span{ϕ1, . . . ,ϕn} over R is an n-dimensional subspace ofC(A) and anyfunction of Xn that hasn distinct zeros inA is identically zero. In that caseXn iscalled aChebyshev spaceor Haar space.

Of course, it is clear thatA in the previous definition must contain at leastnpoints. Usually, we useA = [a,b] andA = T. Since the functions of a Haar systemare linearly independent, we can conclude that any other basis{ψ1, . . . ,ψn} of Xn isalso a Haar system. For many details on these systems see [166] and [42].

Using well-known facts from linear algebra we can formulate a number of equiv-alence:

Proposition 1.2.1. Let H = {ϕ1, . . . ,ϕn} be a Haar system onA. The followingstatements are equivalent:

(a)Eachϕ = a1ϕ1 + · · ·+anϕn 6≡ 0 has at mostn−1 distinct zeros inA.

(b) If x1, . . . ,xn are distinct points ofA, then

D = D(x1, . . . ,xn) =

∣∣∣∣∣∣∣∣

ϕ1(x1) · · · ϕn(x1)...

ϕ1(xn) ϕn(xn)

∣∣∣∣∣∣∣∣6= 0. (1.2.1)

1.2 CHEBYSHEV SYSTEMS AND INTERPOLATION 23

(c) If x1, . . . ,xn are distinct points ofA and f1, . . . , fn are arbitrary numbers, then

there exists a uniqueϕ =n∑

k=1akϕk ∈ Xn = span(H) such that

ϕ(xk) = fk (k = 1, . . . ,n). (1.2.2)

Notice that (1.2.2) is a system of linear equations

a1ϕ1(xk)+ · · ·+anϕn(xk) = fk (k = 1, . . . ,n), (1.2.3)

which has a unique solution for coefficientsa1, . . . ,an. It is equivalent to (1.2.1).We call the elements ofXn polynomialsandϕ in (1.2.2) aninterpolation poly-

nomialwith prescribed valuesfk at the pointsxk. The statement (c) in Proposition1.2.1 means that an interpolation polynomialϕ exists uniquely. The pointsxk arecalledinterpolation nodes.

1.2.2 Algebraic Lagrange interpolation

Take A = [a,b] and ϕk(x) = xk−1 (k = 1, . . . ,n). Then (1.2.1) becomes the well-known Vandermonde determinant

D = V(x1, . . . ,xn) =

∣∣∣∣∣∣∣∣

1 x1 · · · xn−11

...

1 xn xn−1n

∣∣∣∣∣∣∣∣=

n

∏j<k

(xk−x j). (1.2.4)

Assuming the nodes are distinct, we haveD 6= 0 and therefore{1,x, . . . ,xn−1}andPn−1 are a Haar system and Haar space, respectively. There is consequentlya unique solution of (1.2.3), i.e., an unique algebraic interpolation polynomial

ϕ(x) =n∑

k=1akxk−1 ∈ Pn−1. In the case when the prescribed valuesfk at the nodesxk

are the values assumed by a certain functionf , i.e., fk = f (xk), we sayϕ(x) is theLagrange algebraic interpolation polynomialof f and we denote it byLn f (x).

An explicit form of such a polynomial can be achieved by considering the so-calledfundamental Lagrange polynomialsdefined as follows

`n,k(x) :=n

∏ν=1ν 6=k

x−xνxk−xν

, k = 1, . . . ,n. (1.2.5)

or equivalently by

`n,k(x) :=qn(x)

q′n(xk)(x−xk)(1.2.6)

whereqn is a polynomial of degreen whose zeros are the knotsx1, . . . ,xn, i.e.,

qn(x) =n∏

ν=1(x−xν).

Of course, n,k(x) are algebraic polynomials of degreen−1 and


`n,k(xν) ={

1 if ν = k,0 if ν 6= k,

holds for eachν ,k = 1, . . . ,n. Consequently, we can write the Lagrange polynomialLn f (x) in the following form

Ln f (x) =n

∑k=1

`n,k(x) f (xk) (1.2.7)

which is known as theLagrange interpolation formula.

Example 1.2.1.Let λ1 < · · ·< λn. The system of functions{

xλ1, . . . ,xλn}

is a Haarsystem onA = (0,+∞). For distinct pointsx1, . . . ,xn ∈ A the determinant

∣∣∣∣∣∣∣∣

xλ11 · · · xλn

1...

xλ1n xλn

n

∣∣∣∣∣∣∣∣

is different from zero. Such systems of generalized polynomials are known asMuntzsystemsand generalized polynomials as theMuntz polynomials(cf. [42]). Forλ1 = 0this system is a Haar system onA = [0,+∞). For integer exponentsλk = k− 1(k = 1, . . . ,n), it reduces to the standard polynomial system considered before, andit is a Haar system onA = R.

Remark 1.2.1.The Muntz systems can be considered also for the complex se-quences{λ1, . . . ,λn} (cf. [42]), where we have the following definition forxλ :

xλ = eλ logx, x∈ (0,+∞), λ ∈ C,

and the value atx = 0 to be the limit ofxλ asx→ 0 from (0,+∞) whenever thelimits exists.

1.3 INTERPOLATION BY ALGEBRAIC POLYNOMIALS

1.3.1 Representations and computation of interpolation polynomials

In Subsection 1.2.2 we have already introduced a polynomial of minimal degreewhich interpolates a functionf on n fixed pointsx1,x2, . . . ,xn. Such polynomialis known as the Lagrange interpolation polynomial and it was discovered in 1795by Joseph Louis Lagrange. Otherwise, interpolation by polynomials is a very oldsubject in mathematics. The first formula with equally spaces points{xk} was foundin the 1670’s by Isaac Newton.

Defining so-calleddivided differencesof f recursively by

[x1; f ] = f (x1), [x1, . . . ,xk; f ] =[x2, . . . ,xk; f ]− [x1, . . . ,xk−1; f ]

xk−x1,

1.3 INTERPOLATION BY ALGEBRAIC POLYNOMIALS 25

the Newton interpolation formulacan be given also for non-equally spaced nodes{xk} in the form

Ln f (x) = f (x1)+n

∑k=2

(x−x1) · · ·(x−xk−1)[x1, . . . ,xk; f ]. (1.3.1)

The computation of divided differences requiresn(n−1) additions and12 n(n−1)

divisions (cf. Gautschi [123, p. 92]).

Adding another data point(xn+1, f (xn+1)) requires the generation only the nextdifference[x1, . . . ,xn+1; f ] and adding to (1.3.1) the term

(x−x1) · · ·(x−xn)[x1, . . . ,xn+1; f ].

Lagrange interpolation formula (1.2.7) is very attractive for many theoreticalpurposes, but its application for practical computational work is not so efficient,especially when we want to add a new nodexn+1. However, the formula

Ln f (x) =n

∑k=1

`n,k(x) f (xk) (1.3.2)

can be rewritten in a form that makes the Lagrange interpolation formula efficientcomputationally and also allows additional nodes to be added.

Introducing the auxiliary quantities

λ (n)k =

n

∏ν=1ν 6=k

1xk−xν

, k = 1, . . . ,n, (1.3.3)

the fundamental Lagrange polynomials (1.2.5), i.e., (1.2.6), can be expressed in theform

`n,k(x) =λ (n)

k

x−xkqn(x).

Indeed, dividing (1.3.2) throughn∑

ν=1`n,ν(x)≡ 1, we get

Ln f (x) =

n∑

k=1`n,k(x) f (xk)

n∑

ν=1`n,ν(x)

=

n∑

k=1

λ (n)k

x−xkqn(x) f (xk)

n∑

ν=1

λ (n)ν

x−xνqn(x)

,

i.e.,

Ln f (x) =

n∑

k=1

λ (n)k

x−xkf (xk)

n∑

ν=1

λ (n)ν

x−xν

(x 6∈ {x1, . . . ,xn}

). (1.3.4)


This formula is called thebarycentric formula. Comparing (1.3.4) and (1.3.2) weobtain

`n,k(x) =

λ (n)k

x−xk

n∑

ν=1

λ (n)ν

x−xν

, k = 1, . . . ,n.

An efficient algorithm for computing the required quantitiesλ (n)k was suggested

by Werner [354] (see also [123, p. 96] and [124]):

Starting withλ (1)1 = 1, for k = 2, . . . ,n do

λ (k)ν =

λ (k−1)ν

xν −xk(ν = 1, . . . ,k−1);

λ (k)k = −

k−1

∑ν=1

λ (k)ν .

The first set of equations in thek-loop is a direct consequence of (1.3.3), whilethe second equation follows from the identity

1≡k

∑ν=1

`k,ν(x) =k

∑ν=1

λ (k)ν

k

∏i=1i 6=ν

(x−xi)

by comparing the leading coefficients on the left and right side. It givesk∑

ν=1λ (k)

ν = 0.

The previous algorithm requires exactly(n−1)2 additions and12n(n−1) divi-

sions for computing then quantities (1.3.3). In order to include a new data point(xn+1, f (xn+1), we must only extend thek-loop in the previous algorithm throughn+1.

1.3.2 Interpolation array and Lagrange operators

Without loss of generality, we consider the interpolation on the intervalA= [−1,1].Suppose now that we have a continuous functionf : [−1,1]→ R and a sequence ofpolynomials{qn}n∈N (qn ∈ Pn) such that for eachn≥ 1, the polynomialqn has adegree exactlyn andn distinct zerosxn,k (k = 1, . . . ,n) in [−1,1], i.e.,

−1≤ xn,1 < xn,2 < · · ·< xn,n ≤ 1. (1.3.5)

Let X be the corresponding infinite triangular array of these zeros

X :=

x1,1

x2,1 x2,2...

. . .xn,1 xn,2 . . . xn,n

.... ..

. (1.3.6)


Thus, then-th row of this arrayX consists of the zeros of the polynomialqn. Ac-cording to this fact, sometimes we writeQ instead ofX . For example, taking thezeros of the Chebyshev polynomialsTn, we use the notationT for the correspond-ing triangular array.

Now, we can associate to{qn}n∈N, i.e., to the arrayX , a sequence of the La-grange polynomials{Ln(X , f )}n∈N, defined by

Ln(X , f )(xn,k) = Ln(X , f ;xn,k) = f (xn,k), k = 1, . . . ,n.

Note that obviouslyLn(X , f ) ∈ Pn−1 and the indexn in Ln(X , f ) denotes thenumber of knots. The triangularX is called theinterpolation arrayor thesystem ofinterpolation nodes. In the notationLn(X , f )(x) we want to underline the depen-dence on the system of knotsX .

Regarding to (1.2.7) the Lagrange interpolation formula is

Ln(X , f )(x) = Ln(X , f ;x) =n

∑k=1

`n,k(X ;x) f (xk), (1.3.7)

where`n,k(X ,x) = `n,k(x) (∈ Pn−1) are the fundamental Lagrange polynomialsgiven by

`n,k(X ;x):=n

∏ν=1ν 6=k

x−xn,νxn,k−xn,ν

, k = 1, . . . ,n, (1.3.8)

or equivalently by

`n,k(X ;x):=qn(x)

q′n(xn,k)(x−xn,k), k = 1, . . . ,n. (1.3.9)

Usually, for a fixedn, we simplify the notation puttingxk instead ofxn,k and omittingX (as in Subsection 1.2.2).

For a given interpolation arrayX , according to (1.3.7) we define a sequence ofoperatorsLn(X ):C[−1,1] → Pn−1 (n = 1,2, . . .) such thatLn(X ) f = Ln(X , f ).The sequence{Ln(X )}n∈N defines aninterpolatory process. For eachn ∈ N,Ln(X ) is a linear map andLn(X , f ) = f holds whenf is a polynomial of degreeat mostn−1 ( f ∈ Pn−1).

The main question is the convergenceLn(X , f )→ f whenn→ +∞. In otherwords, what kind of the interpolation arrayX provides this convergence. Here, weconsider the spaceC0 = C[−1,1] of all continuous functionsf equipped with theuniform norm

‖ f‖= ‖ f‖∞ = max|x|≤1

| f (x)|

and considerLn(X ) as a map fromC0 into itself, i.e.,Ln(X ):C0 →C0. It is easyto compare theinterpolation error‖ f −Ln(X , f )‖ with the error of best uniformapproximationEn−1( f ), given by

En−1( f ) = minp∈Pn−1

‖ f − p‖= ‖ f −P∗‖,


whereP∗ (∈ Pn−1) is the polynomial of best uniform approximation to the functionf ∈C0. Since, by the Chebyshev alternation theorem, this polynomial interpolatesfin at leastn points, there exists, for eachf ∈C0, an interpolation arrayY for which

‖ f −Ln(Y , f )‖= En−1( f )

goes to zero asn→+∞. However, for the whole classC0, the situation is much lessfavorable.

Taking the polynomial of best approximationP∗ (∈ Pn−1) to f ∈C0, we get

| f (x)−Ln(X , f ;x)| ≤ | f (x)−P∗(x)|+ |P∗(x)−Ln(X , f ;x)|

≤ En−1( f )+ |Ln(X , f −P∗;x)|

≤(

1+n

∑k=1

|`n,k(X ;x)|)

En−1( f ),

i.e.,| f (x)−Ln(X , f ;x)| ≤ (1+λn(X ;x))En−1( f ), (1.3.10)

where we introduced theLebesgue function

λn(X ;x):=n

∑k=1

|`n,k(X ;x)|. (1.3.11)

Using the norm of the operatorLn(X ):C0 →C0, given by

‖Ln(X )‖:= supf∈C0

‖Ln(X , f )‖‖ f‖ = sup

‖ f‖=1‖Ln(X , f )‖, (1.3.12)

it is easy to see that this norm is exactly the norm of the Lebesgue function (1.3.11),and therefore it is called theLebesgue constantand is denoted byΛn(X ). Thus,

Λn(X ) = ‖Ln(X )‖= ‖λn(X ;x)‖= max−1≤x≤1

|λn(X ;x)|. (1.3.13)

Finally, taking the maximum in (1.3.10) over[−1,1], we get

‖ f −Ln(X , f )‖ ≤ (1+Λn(X ))En−1( f ). (1.3.14)

According to (1.3.10) and (1.3.14), the behavior of (1.3.11) and (1.3.13) play animportant role in the study of the convergence of the Lagrange polynomials.

In 1914 Faber [99] proved that

Λn(X )≥ 112

logn, n≥ 1, (1.3.15)

for any interpolation arrayX . Based on this result Faber [99] obtained:


Theorem 1.3.1.For any fixed interpolation arrayX there exists a functionf ∈C0

for which the interpolation polynomialsLn(X , f ) do not converge uniformly tof ,i.e.,

limsupn→+∞

‖Ln(X , f )‖= +∞. (1.3.16)

Grunwald [145, 146] and Marcinkiewicz [201] obtained independently the fol-lowing result:

Theorem 1.3.2. If T is the Chebyshev array of nodes, then there exists a continu-ous functionf0 for which

limsupn→+∞

|Ln(T , f0;x)|= +∞, x∈ [−1,1].

Finally, in 1980 Erdos and Vertesi [96] proved the following statement on diver-gence of Lagrange interpolation processes:

Theorem 1.3.3.For any interpolation arrayX on [−1,1] one can find a functionf ∈C0 such that

limsupn→+∞

‖Ln(X , f )‖= +∞ a.e. in [−1,1].

Moreover, the divergence set is of second category on[−1,1].

A survey about many results in this direction can be found in [88, 349], as wellas in the monograph [324], written by Szabados and Vertesi.

In sequel, we will give some estimates for the Lebesgue constant regarding to theinterpolation arrays. Before of that, we come back to some classical results about in-terpolation error. Also, we give a historical review of some interesting interpolatoryprocesses.

1.3.3 Interpolation error for some classes of functions

In order to be able to estimate theerror of interpolation,

Rn(x) = Rn(X , f ;x) = f (x)−Ln(X , f ;x), (1.3.17)

for any x 6= xk in [−1,1] we need some additional assumptions about the functionf ∈C[−1,1]. Since the nodesxk, k = 1, . . . ,n, are zeros of the polynomialqn andRn(xk) = 0 for eachxk, it is clear that we can write, in general,

Rn(x) = Φ(x; f ,n,X )qn(x),

but Φ(x; f ,n,X ) depends, above all, on the behavior properties of the functionf .Very often, the error of interpolation (1.3.17) is called theremainder term.


The error in the class of continuous-differentiable functions.By restriction theclassC[−1,1] to the class ofn times continuous-differentiable functionsCn[−1,1],the following well-known estimate for (1.3.17) is valid.

Theorem 1.3.4.Let f ∈ Cn[−1,1] and πn(x) = ∏nk=1(x− xk). Then, for eachx ∈

[−1,1], there exists a pointξ = ξ (x) ∈ (−1,1) such that

Rn(x) = f (x)−Ln( f )(x) =f (n)(ξ )

n!πn(x). (1.3.18)

Applying repeatedly Rolle’s theorem, Augustin Cauchy gave an elegant proof of(1.3.18), which can be found today in each standard book of the classical analysis.

An alternative estimate ofRn(x) can be given in terms of divided differences (cf.[140, p. 74])

Rn(x) = f (x)−Ln( f )(x) = πn(x)[x;x1, . . . ,xn; f ]. (1.3.19)

Notice thatπn is a monic polynomial, whose zeros, in fact, are the zeros ofqn,i.e.,πn is the polynomialqn divided by its leading coefficient. The polynomialπn iscalled thenode polynomial.

TakingMn = max|x|≤1

| f (n)(x)| and using (1.3.18) we obtain the following estimate

|Rn(x)|= | f (x)−Ln( f )(x)| ≤ Mn

n!|πn(x)| (−1≤ x≤ 1). (1.3.20)

For an arbitrary system of knotsX , |πn(x)| ≤ 2n holds for all |x| ≤ 1 and weget a very crude estimate

|Rn(x)| ≤ 2nMn

n!(−1≤ x≤ 1).

But, using this estimate for a restricted class of functions and for arbitrary systemsof nodes, we can prove very easily the uniform convergence ofLn(X , f ) to f in[−1,1]. Namely, if we take functionsf having derivatives of any order uniformlybounded, i.e.,

supn

Mn = supn

(max|x|≤1

| f (n)(x)|)

= M < +∞,

we get

supX

max|x|≤1

| f (x)−Ln(X , f ;x)| ≤ 2nMn!

. (1.3.21)

Since the right side in (1.3.21) tends to zero whenn→ +∞, we have the uniformconvergence ofLn(X , f ) to f in [−1,1] for anyX .

According to Theorem 1.1.7, the Chebyshev nodes (1.1.19) are very often agood choice of interpolation nodes. In that case, (1.3.20) becomes

|Rn(x)|= | f (x)−Ln( f )(x)| ≤ Mn

2n−1n!|Tn(x)| (−1≤ x≤ 1).

Such interpolation is called theChebyshev interpolation. The node polynomialswith equally spaced and Chebyshev nodes are displayed in Fig. 1.3.1.


−1 −0.5 0 0.5 10

0.01

0.02

0.03

0.04

0.05equally spaced nodes

−1 −0.5 0 0.5 10

0.01

0.02

0.03

0.04

0.05Chebyshev nodes

Fig. 1.3.1.Graph of|πn(x)| for n = 7, with equally spaced nodes (left) and Chebyshev nodes(right)

The error in the class of analytic functions. For analytic functions we can givean explicit formula for the remainder term

Rn(z) = f (z)−Ln( f )(z),

using the well-known Cauchy residual theorem.

Theorem 1.3.5.LetΓ be a simple closed contour inC and let the nodesxk = xn,k ∈intΓ . For an analytic functionz 7→ f (z) on and insideΓ , we have

Rn(z) =1

2π i

∮

Γ

πn(z)πn(ζ )

f (ζ )ζ −z

dζ (z∈ intΓ ), (1.3.22)

whereπn(z) = ∏nk=1(z−xk).

Proof. Define the functionζ 7→ F(ζ ) by

F(ζ ) =πn(z)πn(ζ )

f (ζ )ζ −z

,

which has the simple poles atz and xk, k = 1, . . . ,n, with residues f (z) andf (xk)πn(xk)/((xk − z)π ′n(xk)) = −`n,k(z) f (xk), k = 1, . . . ,n, respectively. Here,`n,k(z) are the fundamental Lagrange polynomials.

An application of the residual theorem gives

12π i

∮

Γ

F(ζ )dζ = f (z)−n

∑k=1

`n,k(z) f (xk) = f (z)−Ln( f )(z),

i.e., (1.3.22). utRemark 1.3.1.The restriction (1.3.5) for nodesxk is omitted in Theorem 1.3.5. Thenodes can be different complex numbers inside the contourΓ .


1.3.4 Uniform convergence in the class of analytic functions

Interpolation polynomials were widely used in the nineteenth century, but withouta rigorous analysis of their convergence. Perhaps the first significant, but negativeresult, is due to Meray in 1884, in the complex plane. He observed that the inter-polation polynomial for the functionz 7→ f (z) = 1/z at thenth roots of unity isLn( f )(z) = zn−1, and clearly this does not converge tof asn→ +∞, on the unitcircle of off. Of course, in this examplef is not an analytic function in|z| ≤ 1 (be-cause of a pole at0), so that no sequence of polynomials (interpolatory or not) canconverge tof uniformly on the unit circle (cf. [196]).

For some distributions of nodesX and functions analytic in a sufficient largeregion, using Theorem 1.3.5, it is possible to prove the uniform convergence of thecorresponding interpolatory process in a sub-region. In another words, the locationof the interpolation points and the analyticity off play an important role in this sub-ject. The key to convergence is contained in the asymptotic behavior of|πn(z)|1/n,and therefore we investigate convergence supposing the existence of the limit

A(z) = limn→+∞

|πn(z)|1/n = limn→+∞

( n

∏k=1

|z−xn,k|)1/n

(1.3.23)

on certain regions in the complex plane and uniform convergence there.

Theorem 1.3.6.Let conditions of Theorem1.3.5 be satisfied,γ be a simple closedcontour insideΓ , and

δ = minζ∈Γz∈D

|ζ −z|, D = intγ ⊂ intΓ .

If there exist the constantsα,β , 0≤ α < β , such that

(∀z∈ D) A(z)≤ α and (∀ζ ∈ Γ ) A(ζ )≥ β , (1.3.24)

whereA is given by(1.3.23), then the interpolatory process{Ln(X , f )}n∈N con-verges tof uniformly inD.

Proof. Let L = `(Γ ) be the length ofΓ andM = maxζ∈Γ | f (ζ )|. Then, accordingto Theorem 1.3.5 we have

|Rn(z)| ≤ 12π

∮

Γ

|πn(z)||πn(ζ )|

| f (ζ )||ζ −z| |dζ | ≤ ML

2πδ·

maxz∈D

|πn(z)|minζ∈Γ

|πn(ζ )|

for eachz∈ D. Basing on (1.3.23) and using (1.3.24) we can conclude that for asufficiently smallε there exist two integersn1,n2 ∈N such that forn > max(n1,n2)the following inequalities

(∀z∈ D) |πn(z)| ≤ (α + ε)n and (∀ζ ∈ Γ ) |πn(ζ )| ≥ (β − ε)n


hold. Thus,maxz∈D

|πn(z)|minζ∈Γ

|πn(ζ )| ≤ qn and |Rn(z)| ≤ ML2πδ

qn,

whereq = (α + ε)/(β − ε). The last estimate holds uniformly forz∈ D.Taking,ε < (β −α)/2 we see thatq < 1 and, therefore, lim

n→+∞Rn(z) = 0 uni-

formly in D = intΓ . utIn sequel, we will restrict our consideration to nodes distributed in[−1,1] like

(1.3.5), with a given limit distributiondµ(x), and investigate the limitA(z), givenby (1.3.23). Precisely, in terms of potential theory, we are interested in the limit caseof the logarithmic potential

Un(z) =1n

log1

|πn(z)| = log1

|πn(z)|1/n,

induced by the discrete measure that has mass1/n at every zero ofπn.In particular, we will analyzed two cases:

1◦ Uniformly distributed nodes over[−1,1]. For the equally spaced nodes

xn,k =−1+2(k−1)

n−1, k = 1, . . . ,n (n≥ 2).

we havedµ(x) = dx/2 and

U(z) = limn→+∞

Un(z) =12

∫ 1

−1log

1|z−x| dx,

for all zoutside[−1,1]. Indeed, from

e−Un(z) = |πn(z)|1/n = 2n

∏k=1

∣∣∣ z+12

− k−1n−1

∣∣∣1/n

,

i.e.,

Un(z) =− log2− 1n

n

∑k=1

log∣∣∣ z+1

2− k−1

n−1

∣∣∣ ,

it follows (by the definition of the integral)

U(z) =− log2−∫ 1

0log

∣∣∣ z+12

− t∣∣∣ dt =

12

∫ 1

−1log

1|z−x| dx.

With a little work, we get

U(z) = 1− 12

Re{(z+1) log(z+1)− (z−1) log(z−1)

}.

Notice thatU(±1) = 1− log2andU(0) = 1.


In the general case, for a given limit distributiondµ(x), the logarithmic potentialcan be expressed in the form

U(z) =∫ 1

−1log

1|z−x| dµ(x). (1.3.25)

2◦ Arc sine distribution of nodes over[−1,1]. Since the limit measure is givenby dµ(x) = π−1(1−x2)−1/2dx, basing on (1.3.25), we have

U(z) =1π

∫ 1

−1log

1|z−x|

dx√1−x2

.

After some computation (cf. Saff and Totik [303, pp. 45–46]) we get

U(z) = log2∣∣z+√

z2−1∣∣ , (1.3.26)

where√

z2−1 denotes the branch that behaves likeznear infinity. For example, thiscase is appeared for the Chebyshev nodes (see Subsection 1.1.4)

xn,k =−cos(2k−1)π

2n, k = 1, . . . ,n. (1.3.27)

Notice that theminussign in (1.3.27) is taken to provide the arrangement (1.3.5).Then, the potential (1.3.26) can be obtained directly from (1.1.24) and (1.3.23),U(z) = log(2/r), wherer =

∣∣z+√

z2−1∣∣. Notice thatU(z) = log2≈ 0.69. . . , when

z∈ [−1,1] (r = 1).

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1equally spaced nodes

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Chebyshev nodes

Fig. 1.3.2.The equipotential curves for equally spaced nodes (left) and Chebyshev nodes(right)

In Fig. 1.3.2 we display the equipotential curves for the uniformly distributednodes and Chebyshev nodes (arc sine distribution). The potential levelsU(z) = c


were taken between0.05 and 0.65 with an equidistant step0.05. In general, asc increases, the level curves “shrink” towards the interval[−1,1]. Putting γc ={z∈ C |U(z) = c}, it is clear thatintγc2 ⊂ intγc1 for c2 > c1. Of course,A(z) =exp(−U(z)).

The case of arc sine distribution of nodes is much nicer. The curves are ellipsesγc = Er , wherec= log(2/r) < log2, i.e.,r = 2e−c > 1. Thelimit ellipseis just the in-terval[−1,1] for r = 1 (see Subsection 1.1.4), and also[−1,1] = E1⊂ intEr ⊂ intER

for r < R. In this case, according to the previous discussion and Theorem 1.3.6, wecan conclude that the Lagrange interpolatory process{Ln(T , f )}n∈N converges uni-formly on [−1,1] (= E1) if f is analytic on[−1,1], i.e., analytic in any regionintEr

(r > 1), no matter how thin. Precisely, the following result holds:

Theorem 1.3.7. If f is analytic on the closed ellipseD = intEr (r > 1), then theLagrange interpolatory process{Ln(T , f )}n∈N converges uniformly onD.

In the case of equally spaced nodes the equipotential curveγ passing through±1 (ends of the interval[−1,1]) is determined byU(z) = U(±1) = log(e/2), andits interior D by U(z) ≥ log(e/2), i.e., A(z) ≤ α = 2/e. Thus,if f is analytic onD, then the corresponding Lagrange interpolatory process converges uniformly on[−1,1] and, of course, on the complex regionD. However, if f has any singularpoint (6= 0) insideD, it is possible to find an equipotential curveγ∗, determined byU(z) = log(1/A(z)) = c∗, where1− log2< c∗ < 1, so that the functionf be analyticinsideD∗ = intγ∗. In that case, the convergence is only onD∗ and, in particular, onthe corresponding central part(−x∗,x∗) of the interval[−1,1]. Thus, for analyticfunctions on the real line, the interpolation at equally spaced points can be a badidea.

−1 −0.5 0 0.5 1−0.5

0

0.5

1

1.5a = 1

e(n) = 0.000791

−1 −0.5 0 0.5 1−0.5

0

0.5

1

1.5a = 1/4

e(n) = 1.18

Fig. 1.3.3.The Runge’s example forn = 11 equally spaced nodes, whena = 1 (left) anda = 1/4 (right)

The following Runge’s example from 1901 gives an excellent explanation of theprevious fact. Namely, let


fa(x):=1

1+(x/a)2 (x∈ [−1,1])

and the interpolation nodes are equally spaced points in[−1,1]. This function isanalytic on the whole real line, but its continuationz 7→ fa(z) to the complex planedoes have poles at±ia, which can be quite close to the interval[−1,1].

Runge showed that for a sufficiently smalla (e.g.,|a| ≤ 1/5)

e(n) = ‖ fa−Ln( fa)‖= max−1≤x≤1

| fa(x)−Ln( fa)(x)| →+∞, n→+∞.

Precisely, according to the previous investigation, the critical value ofa can be de-termined from the equationU(ia) = 1− log2, which givesa∗ ≈ 0.5255. Thus, for|a|> a∗, e(n)→ 0 asn→+∞. The casesa = 1 anda = 1/4 with n = 11 nodes aredisplayed in Fig. 1.3.3.

For small value of the parametera there existsx∗ (depending ona), such that

limn→+∞

| fa(x)−Ln( fa)(x)|={

0 if |x|< x∗,+∞ if |x|> x∗.

Thus, we have pointwise convergence in the central zone(−x∗,x∗) of the interval[−1,1] and divergence in the lateral zones. The boundx∗ as a function ona is givenin Fig. 1.3.4. For example,x∗ = 0.7942. . . for a = 1/4.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

parameter a

bound x*

Fig. 1.3.4.The boundx∗ as a function of the parametera in the Runge’s example

1.3.5 Bernstein’s example of pointwise divergence

. . .


1.3.6 Lebesgue function and some estimates for the Lebesgue constant

For a given interpolation arrayX , in Subsection 1.3.2 we defined the Lebesguefunctionx 7→ λn(X ;x) and the Lebesgue constantΛn(X ) by (1.3.11) and (1.3.13),respectively, and pointed out to their importance in the convergence of interpolationpolynomials.

We start this section with the formulation of elementary properties of theLebesgue functionλn(X ;x) for an arbitrary arrayX given by (1.3.6) (cf. [199],[46]):

1◦ The functionλn(X ;x) is a piecewise polynomial satisfyingλn(X ;x) ≥ 1andλn(X ;x) = 1 if and only if x = xn,k (k = 1, . . . ,n);

2◦ Between the consecutive nodesxn,k−1 and xn,k (k = 2, . . . ,n) the functionλn(X ;x) has a single maximum, which will be denoted byµk(X ),

µk(X ) = maxxn,k−1≤x≤xn,k

λn(X ;x);

3◦ In the intervals(−1,xn,1) and(xn,n,1) the Lebesgue function is convex andmonotone decreasing and increasing, respectively. Ifxn,1 > −1 and xn,n < 1 thevaluesλn(X ,−1) andλn(X ,1) will be denoted byµ1(X ) andµn+1(X ).

We put

mn(X ) = min1≤k≤n+1

µk(X ) and Mn(X ) = max1≤k≤n+1

µk(X ).

It is clear thatΛn(X ) = Mn(X ).In the sequel we discuss some special interpolation arrays. The cases of equidis-

tant nodes and Chebyshev nodes forn = 7 are presented in Fig. 1.3.5.

−1 −0.5 0 0.5 10

1

2

3

4

5equally spaced nodes

−1 −0.5 0 0.5 10

1

2

3

4

5Chebyshev nodes

Fig. 1.3.5.Graph of the Lebesgue function forn = 7, with equally spaced nodes (left) andChebyshev nodes (right)


Equidistant nodes. For equally spaced nodes on[−1,1],

xn,k =−1+2k−1n−1

, k = 1, . . . ,n,

we denote the corresponding array byE (starting with two nodesx2,1 = −1 andx2,2 = 1). As we mentioned before, such choice is usually bad (see, for instance, theexamples of Runge and Bernstein).

. . .

Chebyshev nodes.Taking the Chebyshev nodesxn,k =−cos(2k− 1)π/(2n), k =1, . . . ,n, Bernstein [30] established an asymptotic behavior ofΛn(T ) in the form

Λn(T )∼2π

logn (n→+∞). (1.3.28)

There are several estimates forΛn(T ). Some of them were obtained by usingthe fact that

Λn(T ) = λn(T ,1) =1n

n

∑k=1

cot(2k−1)π

4n.

For example, Rivlin [296] proved that

a0 +2π

logn < Λn(T ) < 1+2π

logn,

where

a0 =2π

(γ + log

8π

)= 0.9625. . . ,

. . .

Without loss of generality, the mentioned conjecture can be formulate only forcanonical arraysX , i.e., whenxn,1 = −1 andxn,n = 1. Erdos [89, 90, 91] conjec-tured that there is a unique canonical arrayX ∗ for which all maxima in Lebesguefunction are equal, as well as that for any arrayX ,

mn(X ) = min2≤k≤n

µk(X )≤Λn(X ∗)≤ max2≤k≤n

µk(X ) = Mn(X ). (1.3.29)

In 1978 Kilgore [168] and de Boor and Pinkus [40] proved this so-calledBern-stein-Erdos conjecture:

Theorem 1.3.8.Letn≥ 3. Then there exists a unique optimal canonical arrayX ∗such that

µ2(X ∗) = · · ·= µn(X ∗). (1.3.30)

Moreover, for arbitrary arrayX the inequalities(1.3.29) hold.

. . .

2. Orthogonal Polynomials and Weighted PolynomialApproximation

2.1 ORTHOGONAL SYSTEMS AND POLYNOMIALS

2.1.1 Inner product space and orthogonal systems

Suppose thatX is a complex linear space of functions with an inner product( f ,g):X2 → C such that

(a) ( f +g,h) = ( f ,h)+(g,h) (Linearity),

(b) (α f ,g) = α( f ,g) Homogeneity),

(c) ( f ,g) = (g, f ) (Hermitian Symmetry),

(d) ( f , f )≥ 0, ( f , f ) = 0 ⇐⇒ f = 0 (Positivity),

where f ,g,h∈ X andα is a complex scalar. The bar in the above line designates thecomplex conjugate. The spaceX will be called aninner product space.

If X is a real linear space, then the inner product( f ,g):X2 → R is such that thecondition (c) is reduced to

(c′) ( f ,g) = (g, f ) (Symmetry).

A nice inequality for an inner product is known asCauchy-Schwarz-Buniakow-sky inequality(cf. [238, p. 87])

|( f ,g)| ≤ ‖ f‖‖g‖ ( f ,g∈ X), (2.1.1)

where thenormof f is defined by‖ f‖=√

( f , f ).A systemSof elements of an inner product space is calledorthogonal if ( f ,g) =

0 for every f 6= g ( f ,g∈ S). If ( f , f ) = 1 for each f ∈ S, then the system is calledorthonormal.

Suppose thatU = {g0,g1,g2, . . .} is a system of linearly independent functionsin a complex inner product spaceX. Starting from this system of elements andusing the well-knownGram-Schmidt orthogonalizing processwe can construct thecorresponding orthogonal (orthonormal) systemS= {ϕ0,ϕ1,ϕ2, . . .}, whereϕn is,in fact, a linear combination of the functionsg0, g1, . . ., gn, such that(ϕn,ϕk) = 0for n 6= k.

Using the functionsgn andGram matrixof the ordern+1,

40 2. ORTHOGONAL POLYNOMIALS AND WEIGHTED APPROXIMATION

Gn+1 =

(g0,g0) (g0,g1) · · · (g0,gn)

(g1,g0) (g1,g1) (g1,gn)...

(gn,g0) (gn,g1) (gn,gn)

,

a closed expression for the orthogonal functionsϕn can be obtained. Notice that thismatrix is regular. Namely, it is well-known that∆n+1 = detGn+1 6= 0 if and onlyif the system of functions{g0,g1,g2, . . . ,gn} is linearly independent. Moreover, wecan prove that detGn+1 > 0.

Firstly, the matrixGn+1 is Hermitian, because of the property (c) of the innerproduct, i.e.,(gi ,g j) = (g j ,gi). Puttingx = [x0 x1 · · · xn]T andψn = ∑n

k=0 xkgk, wecan see that the Gram matrix is also positive definite. Namely, then

x∗Gn+1x =n

∑i=0

n

∑j=0

(gi ,g j)xix j

can be expressed in the form

x∗Gn+1x = (ψn,ψn) = ‖ψn‖2,

which is positive, exceptψn = 0 (i.e.,x = 0). It means that∆n+1 = detGn+1 > 0.

Theorem 2.1.1.The orthonormal functionsϕn are given by

ϕn(z) =1√

∆n∆n+1

∣∣∣∣∣∣∣∣∣∣

(g0,g0) (g0,g1) · · · (g0,gn−1) g0(z)

(g1,g0) (g1,g1) (g1,gn−1) g1(z)...

(gn,g0) (gn,g1) (gn,gn−1) gn(z)

∣∣∣∣∣∣∣∣∣∣

(2.1.2)

where∆n = detGn and ∆0 = 1.

Proof. For the proof of this statements it is enough to prove thatϕn given by (2.1.2)satisfies the orthogonality condition(ϕn,gk) = 0 for eachk = 0,1, . . . ,n−1.

Since

(ϕn,gk) =1√

∆n∆n+1

∣∣∣∣∣∣∣∣∣∣

(g0,g0) (g0,g1) · · · (g0,gn−1) (g0,gk)

(g1,g0) (g1,g1) (g1,gn−1) (g1,gk)...

(gn,g0) (gn,g1) (gn,gn−1) (gn,gk)

∣∣∣∣∣∣∣∣∣∣

,

we see immediately that this determinant is equal to zero for eachk = 0,1, . . . ,n−1, and fork = n we have(ϕn,gn) =

√∆n+1/∆n. Expanding the determinant from

(2.1.2) along the last column, we obtain an expansion in terms ofgk,

2.1 ORTHOGONAL SYSTEMS AND POLYNOMIALS 41

ϕn(z) =1√

∆n∆n+1

(c0g0(z)+c1g1(z)+ · · ·+cngn(z)

),

wherecn = ∆n. Therefore,

(ϕn,ϕn) =cn√

∆n∆n+1(gn,ϕn) = 1.

ut

2.1.2 Fourier expansion and best approximation

Taking the orthonormal system of functionsS= {ϕ0,ϕ1,ϕ2, . . .}, it is easy to con-struct the correspondingFourier expansionfor a given functionf ∈ X,

f (z)∼+∞

∑k=0

fkϕk(z). (2.1.3)

TheFourier coefficientsfk are given by

fk = ( f ,ϕk) (k = 0,1, . . .), (2.1.4)

which follows directly from (2.1.3). This sequence of coefficients is bounded. In-deed, applying Cauchy-Schwarz-Buniakowsky inequality (2.1.1) to (2.1.4), we ob-tain

| fk|= |( f ,ϕk)| ≤ ‖ f‖‖ϕk‖= ‖ f‖.The partial sums of (2.1.3), i.e.,

sn(z) =n

∑k=0

fkϕk(z), (2.1.5)

play very important role in approximation theory.Let Xn is a subspace ofX spanned bySn = {ϕ0,ϕ1, . . . ,ϕn} (dimXn = n+ 1),

i.e., Xn = spanSn. The following theorem shows thatfn is the closest element tof ∈ X among the all elements of the subspaceXn with respect to the metric inducedby the given norm. Thus, the partial sumsn is thebest approximationto f ∈ X inthe subspaceXn.

Theorem 2.1.2.Let f ∈ X and Xn be a subspace ofX spanned by{ϕ0,ϕ1, . . .}.Then

minϕ∈Xn

‖ f −ϕ‖2 = ‖ f −sn‖2 = ‖ f‖2−n

∑k=0

| fk|2, (2.1.6)

wheresn is given by(2.1.5).


Proof. Let f ∈ X and letsn be given by (2.1.5). An arbitrary element ofXn canbe expressed as a linear combination of orthonormal functionsϕ0,ϕ1, . . . ,ϕn, i.e.,ϕ = ∑n

k=0akϕk. Then

‖ f −ϕ‖2 = ( f −ϕ, f −ϕ) = ( f , f )− ( f ,ϕ)− (ϕ, f )+(ϕ,ϕ).

Since

( f ,ϕ) =n

∑k=0

ak( f ,ϕk) =n

∑k=0

ak fk, (ϕ, f ) =n

∑k=0

ak fk, (ϕ,ϕ) =n

∑k=0

akak,

we get

‖ f −ϕ‖2 = ‖ f‖2−n

∑k=0

| fk|2 +n

∑k=0

(fk fk− ak fk−ak fk +akak

)

= ‖ f‖2−n

∑k=0

| fk|2 +n

∑k=0

| fk−ak|2.

This expression attains the minimal value forak = fk (k = 0,1, . . . ,n), i.e., whenϕ = sn, and the minimum is given by (2.1.6). ut

Since( f ,sn) = (sn,sn) = ∑nk=0 | fk|2, we see that the error in best approximation

en = f −sn is orthogonal tosn, i.e.,

( f −sn,sn) = 0.

Basing on (2.1.6) we conclude that

‖ f −s0‖ ≥ ‖ f −s1‖ ≥ ‖ f −s2‖ ≥ · · · ,which also follows directly from the fact thatX0 ⊂ X1 ⊂ X2 ⊂ ·· ·.

Notice also that (2.1.6) implies theBessel inequality

n

∑k=0

| fk|2 ≤ ‖ f‖2,

which holds for everyn∈ N. Whenn→+∞, it becomes

+∞

∑k=0

| fk|2 ≤ ‖ f‖2.

Thus, the series on the left hand side in this limit inequality converges, which impliesthat

limk→+∞

fk = limk→+∞

( f ,ϕk) = 0.

Therefore, we conclude that the Fourier coefficients of any functionf ∈X approachzero.

The limit Bessel inequality reduces to an equality (Parseval’s equality) in thecase whenspan{ϕ0,ϕ1,ϕ2, . . .} is dense inX.


2.1.3 Examples of orthogonal systems

In this section we mention several interesting orthogonal systems.

Trigonometric system. The trigonometric system{

1, cosx, sinx, cos2x, sin2x, . . . , cosnx, sinnx, . . .}

is orthogonal with respect to the inner product defined by

(u,v) =∫

Tu(x)v(x)dx=

∫ 2π

0u(x)v(x)dx.

The corresponding orthonormal system is

{ 1√2π

,cosx√

π,

sinx√π

,cos2x√

π,

sin2x√π

, . . . ,cosnx√

π,

sinnx√π

, . . .}.

Chebyshev polynomials.Let

( f ,g) =∫ 1

−1f (x)g(x)(1−x2)λ−1/2dx, λ >−1/2. (2.1.7)

Chebyshev polynomials of the first kind{Tn}n∈N0 and of the second kind{Un}n∈N0

are orthogonal on[−1,1] with respect to the inner product (2.1.7) forλ = 0 andλ = 1, respectively (see Subsection 1.1.4). The corresponding orthonormal systemsare

{1√π

,

√2π

T1,

√2π

T2, . . .

}and

{√2π

,

√2π

U1,

√2π

U2, . . .

}, (2.1.8)

respectively.

Orthogonal polynomials on the unit circle. The system of monomials{zn}n∈N0

is orthonormal with respect to the inner product

( f ,g) =1

2π

∫ π

−πf (eiθ )g(eiθ )v(θ)dθ , (2.1.9)

wherev(θ) = 1. This is the simplest case of polynomials orthogonal on the unit cir-cle with respect to (2.1.9). Such polynomials have been introduced and studied bySzego ([326], [327]) and Smirnov ([312], [313]). A more general case was consid-ered by Achieser and Kreın [7], Geronimus ([132], [133]), Nevai ([268], [269]), etc.(see also surveys [9] and [240]). These polynomials are linked with many questionsin the theory of time series, digital filters, statistics, image processing, scatteringtheory, control theory, etc.

Orthogonal polynomials on the unit disk.

Orthogonal polynomials on the ellipse.


Malmquist-Takenaka system of rational functions.

Polynomials orthogonal on the radial rays.

Muntz orthogonal polynomials.

Muntz orthogonal polynomials of the second kind.

Generalized exponential polynomials.

Discrete Chebyshev polynomials.

Formal orthogonal polynomials with respect to a moment functional. (a) Or-thogonality with respect to an oscillatory weight.

(b) Orthogonality on the semicircle.

In this chapter we consider mainly the polynomial orthogonal systems, whenan inner product is defined on some lines or on a curve in the complex planeC.Furthermore, starting from Section 2.2, we consider only polynomials orthogonalon the real line.

By Pn we denote the set of all algebraic polynomials (with complex coefficients)of degree at mostn. Further, letPn be the set of all monic polynomials of degreen,i.e.,

Pn ={

zn +q(z) | q(z) ∈ Pn−1}.

A system of polynomials{pn}, where

pn(z) = γnzn + lower degree terms, γn > 0,

(pn, pm) = δnm, n,m≥ 0,(2.1.10)

is called a system oforthonormal polynomialswith respect to the inner product( · , ·). In many considerations and applications we use themonic orthogonal poly-nomials

πn(z) =pn(z)

γn= zn + lower degree terms. (2.1.11)

Sometimes, we also use the notationpn(z) for monic orthogonal polynomials in-stead ofπn(z).

2.1.4 Basic facts on orthogonal polynomials and extremal problems

Let dµ be a finite positive Borel measure in the complex planeC, with an infiniteset as its support,L2(dµ) denotes the Hilbert space of measurable functionsf forwhich

∫ | f (z)|2dµ(z) < +∞, and let the inner product( · , ·) be defined by

( f ,g) =∫

f (z)g(z)dµ(z) ( f ,g∈ L2(dµ)). (2.1.12)

Starting from the system of monomialsU = {1,z,z2, . . .}, by Gram-Schmidt or-thogonalization process, we can obtain the unique orthonormal polynomials (2.1.10).


In order to emphasize the orthogonality with respect to the given measuredµ, wewrite

pn(z) = pn(dµ;z) = γnzn + lower degree terms, γn = γn(dµ) > 0.

Also, monic orthogonal polynomials (2.1.11) are unique.The following extremal property characterizes orthogonal polynomials:

Theorem 2.1.3.The polynomialπn(z) = pn(z)/γn = zn + · · · is the unique monicpolynomial of degreen of the minimalL2(dµ)-norm, i.e.,

minp∈Pn

∫|p(z)|2dµ(z) =

∫|πn(z)|2dµ(z) =

1γ2n

. (2.1.13)

Proof. Using the polynomials{pk(z)}, orthonormal with respect to the measuredµ,an arbitrary monic polynomialp(z) ∈ Pn can be expressed in the form

p(z) =n−1

∑k=0

ckpk(z)+1γn

pn(z).

Then, we have

‖p‖2 =n−1

∑k=0

|ck|2 +1γ2

n≥ 1

γ2n,

with an equality if and only ifc0 = c1 = · · ·= cn−1 = 0, i.e.,

p(z) = p∗(z) =1γn

pn(z) = πn(z).

utThis extremal property is completely equivalent to orthogonality. Namely, many

questions regarding orthogonal polynomials can be answered by using only thisextremal property (cf. [333] and [315]).

Notice also that the previous theorem gives the polynomial of the best approx-imation to the monomialzn in the classPn−1. Indeed, according to Theorem 2.1.2,the bestL2-approximation tof (z) = zn is given bysn−1(z) = ∑n−1

k=0 fkpk(z), wherefk = (zn, pk), k = 0,1, . . . ,n−1. But, by Theorem 2.1.3,f (z)−sn−1(z) = πn(z), sothat the polynomial of best approximation in this case can be expressed in the formsn−1(z) = zn−πn(z).

Now, we define the function(z, t) 7→ Kn(z, t) by

Kn(z, t) =n

∑k=0

pk(z)pk(t) (n≥ 0), (2.1.14)

which plays a fundamental role in the integral representation of partial sums of theorthogonal expressions.


For a functionf ∈ L2(dµ) we can determine its Fourier coefficientsfk with re-spect to the inner product (2.1.12). Thus, using (2.1.4) and orthonormal polynomials{pk(z)}, we have

fk = ( f , pk) =∫

f (t)pk(t)dµ(t).

Then, the partial sums (2.1.5) can be expressed in an integral form

sn(z) =n

∑k=0

fkpk(z) =n

∑k=0

( f , pk)pk(z) =∫

f (t)Kn(z, t)dµ(t).

Suppose thatf is an arbitrary polynomial of degree at mostn, i.e., f (z) = P(z)(P(z) ∈ Pn). Then, the corresponding partial sumsn coincides withf and we obtain

P(z) =∫

P(t)Kn(z, t)dµ(t) (P(z) ∈ Pn). (2.1.15)

Because of that, the functionKn is called very often thereproducing kernel. NoticethatKn(z, t) = Kn(t,z) and

Kn(z,z) =n

∑k=0

|pk(z)|2 ≥ |p0(z)|2 = γ20 > 0

for eachz∈ C andn≥ 0.The reciprocal of this function is known as theChristoffel function,

λn(z) = λn(dµ;z) =1

Kn−1(z,z)=

(n−1

∑k=0

|pk(z)|2)−1

. (2.1.16)

The following extremal problem is related to the reproducing kernel (cf. [251]and [341]):

Theorem 2.1.4.For everyP(z) ∈ Pn such thatP(t) = 1, we have∫|P(z)|2dµ(z)≥ λn+1(dµ; t), (2.1.17)

with equality only for

P(z) = P∗(z) =Kn(z, t)Kn(t, t)

.

Proof. Let t be a fixed complex number andP(z) ∈ Pn. In order to find the mini-mum of the integral on the left hand side in (2.1.17) under the constraintP(t) = 1,we representP(z) as a linear combination of the orthonormal polynomialspk(z) =pk(dµ;z), i.e.,P(z) = ∑n

k=0ckpk(z). Then, we have

F(P) =∫|P(z)|2dµ(z) =

n

∑k=0

|ck|2.


SinceP(t) = ∑nk=0ckpk(t) = 1, using Cauchy inequality for the complex sequences

c = {ck}nk=0 andp = {pk(t)}n

k=0 (see Mitrinovic [255, p. 32]), we have

1 =

∣∣∣∣∣n

∑k=0

ckpk(t)

∣∣∣∣∣2

≤(

n

∑k=0

|ck|2)(

n

∑k=0

|pk(t)|2)

= F(P)Kn(t, t), (2.1.18)

from which followsF(P)≥ 1/Kn(t, t) = λn+1(dµ; t), i.e., (2.1.17).According to equality case in (2.1.18), which is attained only if the sequencesc

andp are proportional, i.e., whenck = γ pk(t) (k = 0,1, . . . ,n), with some complexconstantλ , we find that

P(t) = γn

∑k=0

|pk(t)|2 = γKn(t, t) = 1.

Thus,γ = 1/Kn(t, t) and the extremal polynomial is given by

P(z) = P∗(z) = γn

∑k=0

pk(t) pk(z) =Kn(z, t)Kn(t, t)

.

utAccording to this theorem, the Christoffel function can be expressed also in the

formλn(dµ ; t) = min

P∈Pn−1P(t)=1

∫|P(z)|2dµ(z). (2.1.19)

Using the previous theorem we can also prove:

Theorem 2.1.5.Let t be a fixed complex constant, and letP(z) be an arbitrarypolynomial of degree at mostn, normalized by the condition

‖P‖2 =∫|P(z)|2dµ(z) = 1.

The maximum of|P(t)|2 taken over all such polynomials is attained for

P(z) = γKn(z, t)√Kn(t, t)

(|γ|= 1).

The maximum itself isKn(t, t).

Proof. Taking Q(z) = P(z)/P(t) we see thatQ(t) = 1 and, according to Theo-rem 2.1.4, ∫

|Q(z)|2dµ(z) =1

|P(t)|2 ≥ λn+1(dµ ; t),

with equality case forQ(z) = P(z)/P(t) = Kn(z, t)/Kn(t, t).Thus,|P(t)|2 ≤ Kn(t, t), with equality only for

P(z) = P(t)Q(z) = γ√

Kn(t, t)Kn(z, t)Kn(t, t)

= γKn(z, t)√Kn(t, t)

,

where|γ|= 1. ut


The extremal property from Theorem 2.1.3, under certain conditions, can beextended toLr(dµ)-norm (1 < r < +∞), so that the unique monic polynomialp∗n(z) = zn + · · · of the minimalLr(dµ)-norm exists, i.e.,

minp∈Pn

∫|p(z)|rdµ(z) =

∫|p∗n(z)|rdµ(z). (2.1.20)

For measures with support on the real line, an interesting special caser = 2s+2,wheres∈N0, leads to a case of thepower orthogonality. Then, the extremal (monic)polynomials in (2.1.20), denoted byp∗n(x) = πn,s(x) = πn,s(x;dµ), exist uniquelyand they are known ass-orthogonal polynomials(for more details see [275], [134],[135], [116], [245]. These polynomials must satisfy the “orthogonality conditions”(cf. Ghizzetti and Ossicini [135], Milovanovic [245])

∫

Rπn,s(x)2s+1xk dµ(x) = 0, k = 0,1, . . . ,n−1. (2.1.21)

In the cases= 0, thes-orthogonal polynomials reduce to the standard orthogonalpolynomials,πn,0 = πn.

Also, thegeneralized Christoffel functioncan be defined for0 < r < +∞, by

λn(dµ, p; t) = minP∈Pn−1P(t)=1

∫|P(z)|r dµ(z). (2.1.22)

Notice that (2.1.22) forr = 2 reduces to (2.1.19), i.e.,λn(dµ ,2;t) = λn(dµ; t).Several properties of the generalized Christoffel functions for measures onR

can be found in Nevai [265, pp. 106–123].

2.1.5 Zeros of orthogonal polynomials

Now, we study some basic properties of the zeros of orthogonal polynomials. Ac-cording to the fundamental theorem of algebra, we know that any polynomial ofdegreen has exactlyn zeros, counting multiplicities. The zeros of orthogonal poly-nomials play very important role in interpolation theory, quadrature formulas, etc.

Using Theorem 2.1.3 it is easy to prove a general result on the location of zeros.This result is connected with the support of the measure supp(dµ), which is a closedset. Firstly, we need some definitions:

Definition 2.1.1. A set A⊂ C is convexif for each pair of pointsz, t ∈ A the lineconnectingzandt is a subset ofA.

Definition 2.1.2. Theconvex hullCo(B) of a setB⊂ C is the smallest convex setcontainingB.

Definition 2.1.3. Let D∞ be the connected component of the complement ofE thatcontains the point∞, thenD∞ is open and

Pc(E) = C\D∞

is thepolynomial convex hullof E.

2.2 ORTHOGONAL POLYNOMIALS ON THE REAL L INE 49

It is clear that Co(supp(dµ)) is the intersection of all closed half-planes con-taining supp(dµ). Also,

supp(dµ)⊂ Pc(supp(dµ))⊂ Co(supp(dµ)).

The following result due to Fejer (see [302] and [341]).

Theorem 2.1.6.All the zeros of the(monic) polynomialπn(dµ;z) lie in the convexhull of the supportE = supp(dµ).

Proof. . . . utAn improvement of this theorem was given by Saff [302]:

Theorem 2.1.7. If Co(supp(dµ)) is not a line segment, then all the zeros of thepolynomialπn(dµ;z) lie in the interior of Co(supp(dµ)).

For example, ifC is the unit circle|z|= 1and supp(dµ)⊂C, then Theorem 2.1.7asserts that all the zeros ofπn(dµ;z) must lie in the open unit disk|z|< 1. This is aclassical result of Szego [328, p. 292] for polynomials orthogonal on the unit circle.

An interesting question is related with a number of zeros ofπn(dµ;z) which areoutsideE = supp(dµ) (i.e., in Co(E)\E). If the setE has holes, it is possible thatall the zeros be in the holes, as in the case of polynomials orthogonal on the unitcircle. Here, we mention a result of Widom [356] (see Saff [302] for the proof).

Theorem 2.1.8.LetE = supp(dµ) andA be a closed set such thatA∩Pc(E) = /0.Then the number of zeros ofπn(dµ;z) onA is uniformly bounded inn.

2.2 ORTHOGONAL POLYNOMIALS ON THE REAL L INE

2.2.1 Basic properties

One of the most important class of orthogonal polynomials is the class of orthogonalpolynomials on the real line. In this subsection we consider the basic facts on suchpolynomials. Thus, we suppose here that the support of a positive Borel measuredµis on the real line, i.e.,

supp(dµ) ={

x∈ R | µ(x− ε,x+ ε) > 0 for every ε > 0},

as well as that all momentsµk =∫R xk dµ(x), k = 0,1, . . ., exist and be finite. Also,

we suppose that supp(dµ) contains infinitely many points, i.e., that the distribu-tion functionµ:R→ R is a non-decreasing function with infinitely many points ofincrease. We are now working with real-valued functions so that the inner product(2.1.12) reduces to

( f ,g) =∫

Rf (x)g(x)dµ(x). (2.2.1)


As before, orthonormal and monic polynomials will be denoted bypn(x) andπn(x),respectively (see (2.1.10) and (2.1.11)). TakingU = {1,x,x2, . . .}, these polynomialscan be obtained by the Gram-Schmidt orthogonalizing process. The correspondingGram matrix can be expressed in terms of the momentsµk (k = 0,1, . . . ,2n) in theform

Gn+1 =

µ0 µ1 · · · µn

µ1 µ2 µn+1...

µn µn+1 µ2n

, (2.2.2)

and the orthonormal polynomial of degreen as

pn(x) = pn(x;dµ) =1√

∆n∆n+1

∣∣∣∣∣∣∣∣∣∣

µ0 µ1 · · · µn−1 1

µ1 µ2 µn x...

µn µn+1 µ2n−1 xn

∣∣∣∣∣∣∣∣∣∣

,

where∆n = detGn and ∆0 = 1. The leading coefficient is given byγn = γn(dµ) =√∆n/∆n+1. The matrices of the form (2.2.2) are known asHankel matricesand the

corresponding determinants asHankel determinants(see (??)).If µ is an absolutely continuous function, then we say thatµ ′(x) = w(x) is a

weight function. In that case, the measuredµ can be express asdµ(x) = w(x)dx,where the weight functionx 7→w(x) is a non-negative and measurable in Lebesgue’ssense for which all moments exists andµ0 > 0. Then, instead ofpn(dµ ; ·), γn(dµ),supp(dµ), . . ., we usually writepn(w; ·), γn(w), supp(w), . . ., respectively. Ifsupp(w) = [a,b], where−∞ < a < b < +∞, we say that{pn} is a system of or-thonormal polynomials in a finite interval[a,b]. For(a,b) we say that it is aninter-val of orthogonality.

The case of adiscrete measure, whendµ(x) is concentrated on a finite numberof points we will not consider here. In the general case, the functionµ can be writtenin the formµ = µac+µs+µj , whereµac is absolutely continuous,µs is singular, andµj is a jump function.

Now we give a few basic properties of orthogonal polynomials on the real line.

Three-term recurrence relation. The following statement gives a fundamental re-lation for polynomials orthogonal with respect to the inner product (2.2.1):

Theorem 2.2.1.The system of orthonormal polynomials{pn(x)}, associated withthe measuredµ(x), satisfy a three-term recurrence relation

xpn(x) = bn+1pn+1(x)+anpn(x)+bnpn−1(x) (n≥ 0), (2.2.3)

wherep−1(x) = 0 and the coefficientsan = an(dµ) andbn = bn(dµ) are given by

an =∫

Rxpn(x)2dµ(x) and bn =

∫

Rxpn−1(x)pn(x)dµ(x) =

γn−1

γn.


Proof. The three-term recurrence relation is a consequence of the inner product(2.1.6) that(x f,g) = ( f ,xg). Indeed, expandingxpn(x) in terms of orthonormalpolynomialspk(x) (k = 0,1, . . . ,n+1),

xpn(x) =n+1

∑k=0

c(n)k pk(x),

where the Fourier coefficients are given by

c(n)k =

∫

Rxpn(x)pk(x)dµ(x) = (xpn, pk) = (pn,xpk) (k = 0,1, . . . ,n+1),

we conclude thatc(n)k = 0 for k+1 < n. Then, putting

c(n)n−1 = (pn,xpn−1) = bn and c(n)

n = (xpn, pn) = an,

we find thatc(n)n+1 = (xpn, pn+1) = bn+1. By comparing the leading coefficients on

both sides of (2.2.3) we getγn = bn+1γn+1, i.e.,bn = γn−1/γn. utSincep0(x) = γ0 = 1/

√µ0 andγn−1 = bnγn we have thatγn = γ0/(b1b2 · · ·bn).Notice thatbn > 0 for eachn.

Conversely, for two given real sequences{an}n∈N0 and{bn}n∈N, wherebn > 0for eachn∈ N, one can construct a sequence of polynomials using the three-termrecurrence relation (2.2.3), starting with initial valuesp−1(x) = 0 and p0(x) = 1.It is well-known by Favard’s theorem (cf. Chihara [51]) that there exists a positivemeasuredσ(x) onR such that

∫

Rpn(x)pm(x)dσ(x) = δnm, n,m≥ 0.

The measuredσ(x) is not unique which depends of the fact whether or not theHamburger moment problem is determined. A sufficient condition for a unique mea-sure is the Carleman’s condition given by∑+∞

n=1(1/bn) = +∞. Evidently, it holds if{bn}n∈N is a bounded sequence.

In many considerations and applications it is appropriate to use the monic or-thogonal polynomialsπn(x) = pn(x)/γn = xn + lower degree terms.

Theorem 2.2.2.The monic orthogonal polynomials{πn(x)} satisfy the followingthree-term recurrence relation

πn+1(x) = (x−αn)πn(x)−βnπn−1(x), n = 0,1,2, . . . , (2.2.4)

whereαn = an andβn = b2n > 0.

Because of orthogonality, we have that

αn =(xπn,πn)(πn,πn)

(n≥ 0), βn =(πn,πn)

(πn−1,πn−1)(n≥ 1).


The coefficientβ0, which multipliesπ−1(x) = 0 in three-term recurrence relation(2.2.4) may be arbitrary. Sometimes, it is convenient to define it byβ0 = µ0 =∫Rdλ (x). Then the norm ofπn can be express in the form

‖πn‖=√

(πn,πn) =√

β0β1 · · ·βn . (2.2.5)

Remark 2.2.1.The recursion coefficientsαn andβn in (2.2.4) can be expressed interms of Hankel determinants

∆n =

∣∣∣∣∣∣∣∣∣

µ0 µ1 · · · µn−1

µ1 µ2 µn...

µn−1 µn µ2n−2

∣∣∣∣∣∣∣∣∣and ∆ ′

n =

∣∣∣∣∣∣∣∣∣

µ0 µ1 · · · µn−2 µn

µ1 µ2 µn−1 µn+1...

µn−1 µn µ2n−3 µ2n−1

∣∣∣∣∣∣∣∣∣,

where∆0 = 1 and∆ ′0 = 0. Namely,

αk =∆ ′

k+1

∆k+1− ∆ ′

k

∆k(k≥ 0), βk =

∆k−1∆k+1

∆2k

(k≥ 1).

Christoffel’s formulae. The functionKn, before defined by (2.1.14), reduces nowto

Kn(x, t) =n

∑k=0

pk(x)pk(t) (n≥ 0), (2.2.6)

Notice thatKn(t,x) = Kn(x, t). Using the three-term recurrence relation (2.2.3) wecan prove:

Theorem 2.2.3.LetKn(x, t) be defined by(2.2.6). Then

Kn(x, t) = bn+1pn+1(x)pn(t)− pn(x)pn+1(t)

x− t, (2.2.7)

wherebn+1 is defined in Theorem2.2.1.

Formula (2.2.7) is known as theChristoffel-Darboux identity.Letting t → x we find the confluent form of (2.2.7),

Kn(x,x) =n

∑k=0

pk(x)2 = bn+1(p′n+1(x)pn(x)− p′n(x)pn+1(x)

). (2.2.8)

The following result is known as Christoffel’s formula (cf. Szego [328, p. 29–30]):

Theorem 2.2.4.Let {pn(x)} be a system of orthonormal polynomials associatedwith the measuredµ(x) on [a,b] and let

ρ(x) = c(x−ξ1)(x−ξ2) . . .(x−ξm), c 6= 0, (2.2.9)


be a nonnegative polynomial[a,b], with distinct zerosξν (ν = 1, . . . ,m) outside(a,b). Then the orthogonal polynomials{qn(x)}, associated with the measuredσ(x) = ρ(x)dµ(x), can be expressed in the form

ρ(x)qn(x) =

∣∣∣∣∣∣∣∣∣

pn(x) pn+1(x) . . . pn+m(x)pn(ξ1) pn+1(ξ1) pn+m(ξ1)

...pn(ξm) pn+1(ξm) pn+m(ξm)

∣∣∣∣∣∣∣∣∣. (2.2.10)

Remark 2.2.2.In case of a zeroξk, of multiplicity s (> 1), the corresponding rowsof (2.2.10) should be replaced by the derivatives of order0,1, . . . ,s−1 of the poly-nomialspn(x), pn+1(x), . . ., pn+m(x) atx = ξk.

Zeros. The three-term recurrence relation (2.2.3) suggests us to study an infinite,symmetric, tridiagonal matrix, known asJacobi matrix,

J = J(dµ) =

a0 b1 Ob1 a1 b2

b2 a2 b3.. .

. . .. . .

O

. (2.2.11)

Under condition both sequences{an}n∈N0 and{bn}n∈N are uniformly bounded,the associated Jacobi matrix (2.2.11) can be understood as a linear operatorJ act-ing on`2, space of all complex square-summable sequences, where the value of theoperatorJ at the vectorx is a product of an infinite vectorx and infinite matrixJ inthe matrix sense. The case when the sequences{an}n∈N0 and{bn}n∈N are not uni-formly bounded, an operator acting on`2 cannot be defined that easily. Additionalproperties of the sequence of orthogonal polynomials are needed in order to be ableto define operator uniquely.

Suppose now that supp(dµ) is bounded and denote by∆(dµ) the smallestclosed interval containing supp(dµ). As a corollary of Theorem 2.1.6 we have thatall zeros ofpn(dµ ;x) (n≥ 1) lie in ∆(dµ). Furthermore, we can prove that they aremutually different.

Theorem 2.2.5.All zeros ofpn(dµ;x), n≥ 1, are real and distinct and are locatedin the interior of the interval∆(dµ).

Proof. Supposepn(dµ ;x) hasmdistinct zerosx1, . . ., xm in the interior of the inter-val ∆(dµ) that are of odd order and letq(x):= (x−x1) · · ·(x−xm). Then, for eachx∈ ∆(dµ), we have thatpn(dµ;x)q(x)≥ 0, from which follows

∫

Rpn(x)q(x)dµ(x) > 0.

On the other side, ifm< n, because of orthogonality, this integral is equal to zero.This contradiction implies thatm= n, which means that all zeros are simple and arelocated in the interior of the interval∆(dµ). ut


Let xn,k, k = 1, . . . ,n, denote the zeros ofpn(dµ;x) in increasing order

xn,1 < xn,2 < · · ·< xn,n.

Theorem 2.2.6.The zeros ofpn(dµ;x) and pn+1(dµ;x) interlace, i.e.,

xn+1,k < xn,k < xn+1,k+1 (k = 1, . . . ,n; n∈ N).

The proof of this interlacing property can be given using the inequality

p′n+1(x)pn(x)− p′n(x)pn+1(x) > 0 (x∈ R),

which follows from (2.2.8) (cf. [238, p. 105]).Consider now the three-term-recurrence relation (2.2.3), in whichn is substi-

tuted byk. Then, taking this relation fork = 0,1, . . . ,n−1, one can obtain the fol-lowing system of equations

xpn(x) = Jn(dµ)pn(x)+bnpn(x)en,

where

Jn(dµ) =

a0 b1 Ob1 a1 b2

b2 a2. ..

. ... .. bn−1

O bn−1 an−1

, pn(x) =

p0(x)p1(x)p2(x)

...pn−1(x)

, en =

0

0

0...1

.

The tridiagonal matrixJn = Jn(dµ) is then×n leading principal minor matrix ofthe infinity Jacobi matrix (2.2.11). It is clear thatpn(x) = 0 if and only if

xpn(x) = Jnpn(x),

i.e., the zerosxn,k of pn(x) are the same as the eigenvalues of the Jacobi matrixJn. Also, notice that the monic polynomialπn(x) can be expressed in the followingdeterminant form

πn(x) = det(xIn−Jn),

whereIn is the identity matrix of the ordern.Let xi = xn,i , i = 1, . . . ,n, be zeros of the orthonormal polynomialpn(dµ ;x).

Puttingx = xi andt = x j in (2.2.6) we get

Kn(xi ,x j) =n−1

∑k=0

pk(xi)pk(x j) = 〈pn(xi),pn(xi)〉= pn(x j)Tpn(xi), (2.2.12)

wherepn(x) is the vector defined before. The〈 ·, · 〉 is the usual inner product in theEuclidean spaceRn.

2.3 CLASSICAL ORTHOGONAL POLYNOMIALS 55

Theorem 2.2.7.For inner products in(2.2.12) we have

Kn(xi ,x j) =

{0, if i 6= j,

bnpn−1(xi)p′n(xi), if i = j,(2.2.13)

wherebn is defined in Theorem2.2.1.

Proof. . . . utAn important question in interpolation theory is a distance between consecutive

zeros of the orthonormal polynomialpn(dµ;x).

Some special weights.. . .At the end of this subsection we mention some results for polynomials{qn(x)}

orthogonal with respect to an even weight functionx 7→w(x) on a symmetric interval[−a,a]. At first, we have thatqn(−x) = (−1)nqn(x), i.e., qn(x) is an even or oddpolynomial depending on the parity ofn.

Theorem 2.2.8.Let {qn(x)}n∈N0 be a system of polynomials orthogonal with re-spect to the even weightw(x) on (−a,a). Then,

(a){

q2n(√

t)}

n∈N0is a system of polynomials orthogonal on[0,a2] with respect

to the weightw0(t) = w(√

t)/√

t;(b)

{q2n+1(

√t)/√

t}

n∈N0is a system of polynomials orthogonal on[0,a2] with

respect to the weightw1(t) =√

t w(√

t).

Proof. (a) Letn 6= k. Since∫ a

−aq2n(x)q2k(x)w(x)dx= 2

∫ a

0q2n(x)q2k(x)w(x)dx= 0,

by a change of variablesx2 = t, we have

∫ a2

0q2n(

√t)q2k(

√t)

w(√

t)√t

dt = 0 (n 6= k).

Thus, the system of polynomials{

q2n(√

t)}

n∈N0is orthogonal on[0,a2] with respect

to the weightw0(t) = w(√

t)/√

t.The proof of the statement (b) is analogous. ut

2.3 CLASSICAL ORTHOGONAL POLYNOMIALS

2.3.1 Definition of the classical orthogonal polynomials

A survey on characterization theorems for orthogonal polynomials on the real linewas given by Al-Salam [10]. The most important orthogonal polynomials on thereal line are so-called thevery classical orthogonal polynomials(cf. Van Assche


[341], Nikiforov and Uvarov [271], Suetin [319]). An extension of the very classi-cal orthogonal polynomials using difference operators andq-difference operators isknown nowadays as the classical orthogonal polynomials (see Andrews and Askey[12], Andrews, Askey, and Roy [13], Askey and Wilson [18], Atakishiyev andSuslov [19]). Such a much larger class of orthogonal polynomials can be arrangedin a table, which is known as the Askey table and itsq-extension (cf. Koekoek andSwarttouw [172]).

In this subsection we consider onlyvery classical orthogonal polynomials. Inthe sequel we will omit the term “very” and we call such polynomials theclassicalorthogonal polynomials. They are distinguished by several particular properties.

Let the inner product is given by

( f ,g)w =∫ b

af (x)g(x)w(x)dx. (2.3.1)

Since every interval(a,b) can be transformed by a linear transformation to one offollowing intervals:(−1,1), (0,+∞), (−∞,+∞), we will restrict our consideration(without loss of generality) only to these three intervals.

Definition 2.3.1. The orthogonal polynomials{Qn(x)} on (a,b) with respect to theinner product(2.3.1) are called theclassical orthogonal polynomialsif their weightfunctionsx 7→ w(x) satisfy the differential equation

ddx

(A(x)w(x)) = B(x)w(x), (2.3.2)

where

A(x) =

1−x2, if (a,b) = (−1,1),

x, if (a,b) = (0,+∞),

1, if (a,b) = (−∞,+∞),

andB(x) is a polynomial of the first degree. For such classical weights we will writew∈CW.

We note that ifw∈CW, thenw∈C1(a,b), and also the following property:

Theorem 2.3.1. If w∈CW then for eachm= 0,1, . . . we have

limx→a+

xmA(x)w(x) = 0 and limx→b−

xmA(x)w(x) = 0. (2.3.3)

According to the above definition, this class of orthogonal polynomials{Qn(x)}on (a,b) can be classified as

1◦ theJacobi polynomialsP(α,β )n (x) (α,β >−1) on (−1,1);

2◦ thegeneralized Laguerre polynomialsLαn (x) (α >−1) on (0,+∞);

3◦ theHermite polynomialsHn(x) on (−∞,+∞).


Table 2.3.1.The classification of the classical orthogonal polynomials

(a,b) w(x) A(x) B(x) λn

(−1,1) (1−x)α (1+x)β 1−x2 β −α− (α +β +2)x n(n+α +β +1)(0,+∞) xα e−x x α +1−x n

(−∞,+∞) e−x21 −2x 2n

Their weight functions and the corresponding polynomialsA(x) andB(x) aregiven in the following table.

Special cases of the Jacobi polynomials are:

– theGegenbauer polynomialsCλn (x) (α = β = λ −1/2);

– theLegendre polynomialsPn(x) (α = β = 0);

– theChebyshev polynomials of the first kindTn(x) (α = β =−1/2);

– theChebyshev polynomials of the second kindUn(x) (α = β = 1/2);

– theChebyshev polynomials of the third kindVn(x) (α =−β =−1/2);

– theChebyshev polynomials of the fourth kindWn(x) (α =−β = 1/2).

If α = 0, the generalized Laguerre polynomials reduces to thestandard LaguerrepolynomialsLn(x).

The Chebyshev polynomials of the first and second kind were already introducedand studied in Subsections 1.1.2 and 1.1.4. Puttingx = cosθ , −1≤ x≤ 1, thesepolynomials can be expressed in the form (cf. Subsection 1.1.2)

Tn(x) = Tn(cosθ) = cosnθ and Un(x) = Un(cosθ) =sin(n+1)θ

sinθ,

respectively. Similarly, for the Chebyshev polynomials of the third and fourth kindthe following expressions

Vn(cosθ) =cos(n+1/2)θ

cosθ/2and Wn(cosθ) =

sin(n+1/2)θsinθ/2

hold. Notice thatWn(−x) = (−1)nVn(x).There are many characterizations of the classical orthogonal polynomials.Similarly to the well-known inequalities of Landau type [182] and Kolmogoroff

type [173] for continuously-differentiable functions, as well as their generalizations(see, for example, [77], [142], [157], [173], [234], [305], and [317]), it is possi-ble to consider such kind of inequalities for algebraic polynomials of fixed degree(cf. Varma [344], Bojanov and Varma [39], Alves and Dimitrov [11], Agarwal andMilovanovic [3], [4]).

The following characterization of the classical orthogonal polynomials wasgiven by Agarwal and Milovanovic [3]:


Theorem 2.3.2.For all P(x) ∈ Pn the inequality

(2λn +B′(0))‖√

AP′‖2 ≤ ‖AP′′‖2 +λ 2n‖P‖2 (2.3.4)

holds, with equality if only ifP(x) = cQn(x), whereQn(x) is the classical orthogonalpolynomial of degreen orthogonal to all polynomials of degree≤ n−1 with respectto the weight functionw(x) on (a,b), andc is an arbitrary real constant. Theλn,A(x), andB(x) are given in Table 2.3.1.

We mention some special cases.First, for w(x) = e−x2

on (−∞,+∞), the inequality (2.3.4) reduces to Varma’sinequality

‖P′‖2 ≤ 12(2n−1)

‖P′′‖2 +2n2

2n−1‖P‖2 (P(x) ∈ Pn),

which reduces to an equality if and only ifP(x) = cHn(x), whereHn(x) is the Her-mite polynomial of degreen andc is an arbitrary real constant.

In the generalized Laguerre case, the inequality (2.3.4) becomes

‖√xP′‖2 ≤ n2

2n−1‖P‖2 +

12n−1

‖xP′′‖2,

wherew(x) = xαe−x (α >−1) on (0,+∞).In the Jacobi case we get the inequality

((2n−1)(α +β ) + 2(n2 +n−1)

)‖√

1−x2P′‖2

≤ n2(n+α +β +1)2‖P‖2 +‖(1−x2)P′′‖2.

wherew(x) = (1−x)α(1+x)β (α ,β >−1) on (−1,1).Weighted polynomial inequalities inL2-norm of Markov-Bernstein type, as well

as the corresponding connections with the classical orthogonal polynomials, wereconsidered in [149], [150], [147], [148].

A characterization of classical orthogonal polynomials based on the concept of“reversed” continued fraction of Stieltjes type was proposed by Dette and Studden[70]. Using a concept of the dual orthogonal polynomials introduced by de Boorand Saff [41], Vinet and Zhedanov [351] studied their properties and presented alsoa characterization of the classical and semi-classical orthogonal polynomials.

In sequel we give the basic common properties of the classical orthogonal poly-nomials (cf. [4, 240, 251]).

2.3.2 General properties of the classical orthogonal polynomials

Under notation from the previous subsection, we have:

Theorem 2.3.3.The derivatives of the classical orthogonal polynomials{Qn}n∈N0

form also a sequence of the classical orthogonal polynomials.


Proof. Put

Im,n =∫ b

axm−1B(x)Qn(x)w(x)dx=

(xm−1B(x),Qn

)w (m∈ N, n∈ N0).

For eachn > m (= deg(xm−1B(x)

), because of orthogonality, we haveIm,n = 0.

On the other side, using (2.3.2), (2.3.3) and the integration by parts, we obtain

Im,n =∫ b

axm−1Qn(x)

ddx

(A(x)w(x)

)dx

= −(m−1)(xm−2A(x),Qn

)w−

(xm−1A(x),Q′

n

)w.

Since(xm−2A(x),Qn

)w = 0 (m< n), we conclude that

(xm−1A(x),Q′

n

)w =

(xm−1,Q′

n

)Aw = 0,

i.e., the sequence of polynomials{Q′n}n∈N is orthogonal with respect to the weight

functionx 7→w1(x) = A(x)w(x). If w1∈CW, these orthogonal polynomials are clas-sical. Indeed,

ddx

(A(x)w1(x)

)= A′(x)w1(x)+A(x)

ddx

(A(x)w(x)

)

=(A′(x)+B(x)

)A(x)w(x) = B1(x)w1(x),

whereB1(x) = A′(x)+B(x) is a polynomial of the first degree. utApplying the induction method we can prove a more general result:

Theorem 2.3.4.The sequence{Q(m)n }n=m,m+1,... is a classical orthogonal polyno-

mial sequence on(a,b) with respect to the weight functionx 7→wm(x) = A(x)mw(x).The differential equation for this weight is

(A(x)wm(x))′ = Bm(x)wm(x),

whereBm(x) = mA′(x)+B(x).

Theorem 2.3.5.The classical orthogonal polynomialQn(x) is a particular solutionof the second order linear differential equation of hypergeometric type

L[y] = A(x)y′′+B(x)y′+λny = 0, (2.3.5)

where

λn =−n

(12(n−1)A′′(0)+B′(0)

). (2.3.6)


Proof. Let m < n. Since, by Theorem 2.3.3,Im,n =(Q′

n,xm−1

)Aw = 0, using the

integration by parts it reduces to

Im,n =− 1m

∫ b

a

ddx

(A(x)Q′

n(x)w(x))xmdx=− 1

m

(Qn,x

m)w,

where we putQn(x) = A(x)Q′′n(x) + B(x)Q′

n(x). This means that the polynomialQn(x) is orthogonal toPn−1 with respect to the inner product(·, ·)w.

Thus,Qn(x) must be equal toQn(x) up to a multiplicative constant, i.e.,

A(x)Q′′n(x)+B(x)Q′

n(x)+λnQn(x) = 0.

Comparing the coefficients in this equality we getλn in the form (2.3.6). utThe equation (2.3.5) can be written in the Sturm-Liouville form

ddx

(A(x)w(x)

dydx

)+λnw(x)y = 0. (2.3.7)

The coefficientsλn are also displayed in Table 2.3.1.Similarly, them-th derivative ofQn satisfies the differential equation

ddx

(A(x)wm(x)

dydx

)+λn,mwm(x)y = 0, (2.3.8)

whereλn,m = −(n−m)(

12(n+m−1)A′′(0)+B′(0)

). We note that this expression

for λn,m reduces to (2.3.6) form= 0, i.e.,λn,0 = λn.

Remark 2.3.1.The characterization of the classical orthogonal polynomials by dif-ferential equation (2.3.5), i.e. (2.3.7), was proved by Lesky [187], and conjecturedby Aczel [2] (see also Bochner [38]). Such a differential equation appears in manymathematical models in atomic physics, electrodynamics and acoustics. As an ex-ample we mention the well-known Schrodinger equation.

The classical orthogonal polynomials possess a Rodrigues’ type formula (cf.Bateman and Erdelyi [26], Tricomi [335], and Suetin [319]), which can be derivedby the successive application of (2.3.8)n times.

Theorem 2.3.6.The classical orthogonal polynomialQn(x) can be expressed in theform

Qn(x) =Cn

w(x)· dn

dxn

(A(x)nw(x)

), (2.3.9)

whereCn are constants different from zero.

Using the Cauchy formula forn-th derivative of a regular function, (2.3.9) canbe represented in the following integral form


Qn(x) =Cn

w(x)· n!2π i

∮

Γ

A(z)nw(z)(z−x)n+1 dz, (2.3.10)

whereΓ is a closed contour such thatx∈ intΓ .The constantsCn in (2.3.9) and (2.3.10) can be chosen in different way (for

example,Qn to be monic, orthonormal, etc.). A historical reason leads to

Cn =

(−1)n

2nn!for P(α,β )

n (x),

1n!

for Lsn(x),

(−1)n for Hn(x).

In addition, the Gegenbauer and the Chebyshev polynomials need

Cλn (x) =

(2λ )n(λ + 1

2

)n

P(α,α)n (x) (α = λ −1/2), (2.3.11)

Tn(x) =n!(12

)n

P(−1/2,−1/2)n (x),

Un(x) =(n+1)!(

32

)n

P(1/2,1/2)n (x),

where(s)n is the standard notation for Pochhammer’s symbol

(s)n = s(s+1) · · ·(s+n−1) =Γ (s+n)

Γ (s)(Γ is the gamma function).

For such defined polynomials

Qn(x) = kn(xn + rnxn−1 + · · ·), (2.3.12)

using (2.3.9) and the integration by parts, we can get the following formula for thenorm of polynomials (cf. [238, p. 126])

‖Qn‖2 = (Qn,Qn)w = knCn(−1)nn!∫ b

aA(x)nw(x)dx. (2.3.13)

Also, using the Rodrigues’s formula (2.3.9), the leading coefficientkn in Qn(x), aswell as the coefficientrn, can be derived (cf. [238, p. XXX])

kn = Cn

n

∏ν=1

(B′(0)+

12(2n−ν−1)A′′(0)

), (2.3.14)

rn = nB(0)+(n−1)A′(0)B′(0)+(n−1)A′′(0)

. (2.3.15)


By Qn(x) we denote the corresponding monic classical orthogonal polynomials,i.e., Qn(x) = k−1

n Qn(x). According to the recurrence relation (2.2.4) for the monicpolynomials, with recursion coefficientsαn andβn, we conclude that the polynomi-als{Qn(x)} satisfy the recurrence relation

Qn+1(x) =kn+1

kn(x−αn)Qn(x)− kn+1

kn−1βnQn−1(x) (n≥ 0), (2.3.16)

where the leading coefficientskn are given in (2.3.14).In the case of classical orthogonal polynomials one can also expressQ′

n(x) interms ofQn(x) andQn−1(x). Namely,

A(x)Q′n(x) = (enx+ fn)Qn(x)+gnQn−1(x), (2.3.17)

where

en =12

nA′′(0), fn = nA′(0)− 12

rnA′′(0),

gn = =−knβn

kn−1

[B′(0)+

(n− 1

2

)A′′(0)

].

According to the three-term recurrence relation (2.3.16), (2.3.17) can be alsoexpressed in the form

A(x)Q′n(x) = unQn+1(x)+vnQn(x)+wnQn−1(x), (2.3.18)

with the corresponding coefficientsun,vn,wn. Such kind of relation can be taken asa characterization of the classical orthogonal polynomials, supposing thatA(x) is apolynomial of degree not exceeding 2 (cf. [204] and [351]).

The classical polynomialQn(x) can be expressed in terms ofQ′n−1(x), Q′

n(x),andQ′

n+1(x) in the following way

ωnQn(x) = ξnQ′n−1(x)+ηnQ′

n(x)+ζnQ′n+1(x), (2.3.19)

where

ωn = (n+1)mn, mn = B′(0)+12(n−2)A′′(0), ξn =− (n+1)knβn

2kn−1A′′(0),

ζn = =knmn

kn+1, ηn = B(0)+(n−1)A′(0)− 1

2rnA′′(0)− (rn+1− rn)mn.

For some particular cases of the classical orthogonal polynomials, these coeffi-cients will be done later.

2.3.3 Generating function

The classical orthogonal polynomials can be considered as coefficients in Taylorseries of certain analytic functions.


Definition 2.3.2. For a function(x, t) 7→Φ(x, t) we say that thegenerating functionof the system of polynomials{Qk}k∈N0 if, for sufficiently smallt,

Φ(x, t) =+∞

∑k=0

Qk(x)k!

tk ,

whereQk(x) = Qk(x)/Ck andCk is the normalized constant which appears in theRodrigues’ formula (2.3.9).

According to (2.3.10), i.e.,

Qk(x)k!

=1

w(x)· 12π i

∮

Γ

A(z)kw(z)(z−x)k+1 dz,

whereΓ is a closed contour such thatx∈ intΓ , we have

Φ(x, t) =1

2π i· 1w(x)

∮

Γ

w(z)z−x

(+∞

∑k=0

(A(z)tz−x

)k)

dz.

Since for sufficiently smallt,∣∣∣ A(z)t

z−x

∣∣∣< 1, we have

Φ(x, t) =1

2π i· 1w(x)

∮

Γ

w(z)z−x−A(z)t

dt .

If t → 0, we conclude that the equation

z−x−A(z)t = 0 (2.3.20)

has a rootz→ x, and the second root, if there exists, tends to∞. Thus, for a suffi-ciently smallt we can take that the contourΓ enclosing only one rootz= g(x, t),which means that the integrand has only one simple polez= g(x, t) inside the con-tourΓ . Then

Φ(x, t) =1

w(x)Res

z=g(x,t)

{w(z)

z−x−A(z)t

}=

1w(x)

· w(z)1−A′(z)t

∣∣∣∣z=g(x,t)

.

wherez = g(x, t) is the root of the equation (2.3.20), close to the pointz = x forsufficiently smallt.

Example 2.3.1.In the case of Legendre polynomials we havew(x) = 1 andA(x) =1−x2. Then, the equation (2.3.20) reduces toz−x− (

1−z2)t = 0, and we have

g(x, t) =−1+

√1+4t(t−x)2t

,

and then


Φ(x, t) =+∞

∑k=0

Pk(x)Ckk!

tk =1

1+2zt

∣∣∣∣z=g(x,t)

=1√

1+4tx+4t2.

SinceCk = (−1)k/(2kk!), we have

1√1+4tx+4t2

=+∞

∑k=0

Pk(x)(−2t)k ,

i.e.,1√

1−2tx+ t2=

+∞

∑k=0

Pk(x)tk . (2.3.21)

In a similar way as in the previous example, we can get the generating functionfor Jacobi polynomials,

Φ(x, t) =2α+β

R(1− t +R)α(1+ t +R)β =+∞

∑k=0

P(α,β )k (x)tk , (2.3.22)

whereR=√

1−2tx+ t2. Takingα = β = λ −1/2 and using (2.3.11), the generat-ing function (2.3.22) becomes

2λ−1/2

R(1−xt+R)λ−1/2=

+∞

∑k=0

(λ + 1

2

)k

(2λ )kCλ

k (x)tk.

On the other hand, for Gegenbauer polynomials, there is another simpler generatingfunction

(1−2tx+ t2)−λ =+∞

∑k=0

Cλk (x)tk . (2.3.23)

Notice that forλ = 1/2 both those generating functions reduce to (2.3.21).

Example 2.3.2.For the generalized Laguerre polynomials we havew(x) = xαe−x

(α >−1) andA(x) = x. Fromz−x−zt = 0 it follows g(x, t) = x/(1− t), and then

Φ(x, t) =1

xαe−x

(x

1− t

)αe−x/(1−t) · 1

1− t= (1− t)−(α+1)e−xt/(1−t) .

Thus,

(1− t)−(α+1)e−xt/(1−t) =+∞

∑k=0

Lαk (x)tk .

Example 2.3.3.For the Hermite polynomials we havew(x) = e−x2andA(x) = 1.

The equation (2.3.20), in this case, becomesz−x−t = 0, with only one rootg(x, t)=x+ t. According to the previous, we get

Φ(x, t) = e−(x+t)2+x2= e−2xt−t2 .


Thus, we have

e−2xt−t2 =+∞

∑k=0

Hk(x)(−1)kk!

tk ,

i.e.,

e2xt−t2 =+∞

∑k=0

Hk(x)k!

tk . (2.3.24)

Putting in (2.3.23)x/√

λ andt/√

λ (λ > 0) instead ofx andt, respectively, andobserving that

limλ→+∞

(1−2

xtλ

+t2

λ

)= e2xt−t2,

according to (2.3.24), we conclude that

limλ→+∞

λ−k/2Cλk (x/

√λ ) =

Hk(x)k!

.

This means that the Hermite polynomials are limits of Gegenbauer polynomials.Also, it can be proved (cf. [13, p. 306])

limβ→+∞

P(α,β )k (1−2x/β ) = Lα

k (x),

66 References

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