Constr. Approx. (1996) 12:409-422 CONSTRUCTIVE APPROXIMATION �9 1996 Springer-Verlag New York Inc.

Convex Polynomial and Spline Approximation in Lp, 0 < p < oo

R. A. DeVore, Y. K. Hu, and D. Leviatan

Abstract. We prove that a convex function f 6 L p [ -1 , 1], 0 < p < cx~, can be approximated by convex polynomials with an error not exceeding Cw~ (f, 1/n)p where w~(f, .) is the Ditzian-Totik modulus of smoothness of order three of f . We are thus filling the gap between previously known estimates involving og~(f, I/n)p, and the impossibility of having such estimates involving ~o4. We also give similar estimates for the approximation of f by convex C O and C I piecewise quadratics as well as convex C 2 piecewise cubic polynomials.

1. Introduction and Main Results

Let f E Lp[-1, 1], 0 < p < cx~ (where by L~ we mean C[ -1 , 1])be a convex function. We are interested in estimating the degree of approximation of f in the Lp-(quasi-)norm by means of convex polynomials or convex splines.

The first estimates of this type are due to Svedov [SVE] who proved that for a given convex f ~ Lp[-1, 1], 0 < p < oo, and n >_ 2, there exist convex polynomials Pn of degree not exceeding n, such that

(1.1) IIf - Pnl[p < C092(f , 1/n)p,

where C ---- C(p) is an absolute constant independent of f and n, and Coz(f, -)p is the ordinary second-order modulus of smoothness in the Lp-norrn. Svedov [SVE] went on to prove that in (1.1), 092 cannot be replaced by 094 while keeping the constant independent of f and n.

In recent years, (1.1) has been improved in a sequence of papers by DeVore, Leviatan, and Yu (see IDLE], [LY], and [Y]) who were able to replace 09 2 by the Ditzian-Totik second modulus of smoothness in Lp. (See [DIT] for the definition and properties of the latter.) Namely, they proved that for a convex f E Lp[-1, 1], 0 < p < ~ , and each n > 1, there exist convex polynomials Pn of degree < n such that

(1.2) [If -- Pnllp < C09~ (f, 1/n)p.

Date received: February 3, 1995. Date revised: June 30, 1995. Communicated by Dietrich Braess. AMS classification: 41A10; 41A15, 41A17, 41A25, 41A29. Key words and phrases: Degree of convex approximation, Constrained approximation in Lp space, Spline approximation, Polynomial approximation.

409

410 R.A. DeVore, Y. K. Hu, and D. Leviatan

While (1.2) is also valid for p = oe, better estimates are now known in that case. In fact, for a convex f 6 C [ - 1 , 1] there exist convex polynomials pn of degree _< n, such that

(1.3) 1 [ / - Pn[lct-l,ll < C w ~ ( f , 1 / n ) ~ .

(See [KOP] and [HLY] for a weaker but earlier result.) One of the main aims of the present paper is to show that a similar estimate to (1.3) involving co ~ ( f , .)p, 0 < p < c~, v 3 is also valid, thus completely closing the gap left by Svedov [SVE]. To this end, we prove the following result in Section 3.

The o rem 1.1. Let f c L p [ - 1 , 1], 0 < p < ~ , be convex. Then f o r each n > 2, there is a convex polynomial Pn o f degree < n, such that

(1.4) I l f -- Pnllp <-- Cog~ ( f , 1/n)p,

where C is a constant which depends at most on p when p ~ O.

Our proof of Theorem 1.1, given in Section 3, is based inpart on a convex, C O piecewise polynomial approximation to f which is constructed in Section 2.

In Section 4, we consider piecewise polynomial approximation a little further. To describe these results, we let E ( I ) := E ( f , I)p denote the error in Lp approximation of f on the interval I by quadratic polynomials. We recall that from Whitney's theorem [DLO, pp. 182 and 374], we have

(1.5) E ( I ) < Ca)3(f, Ill, l )p,

where Ill denotes the length of the interval I , for all intervals I , with a constant C depending only on p as p --+ 0.

Now, let Tn := { -1 =: to < tl < . . . < tn := 1},n > 1, be any partition of [ - 1 , 1], and set tj := - 1 , j < 0, and tj := 1, j > n. For j = - 1 . . . . . n, l e t / j := [tj, tj+l],

/ (1) : = [ t j - l l , t j + l l ] , and//(2) : = [ t j -49 , t j+49] . Then we have:

T h e o r e m 1.2. Let f c L p [ - 1 , 1], 0 < p < oc, be convex. Then there exists a convex C 1 piecewise quadratic polynomial SO~ and a convex C 2 piecewise cubic polynomial

S(2) on the partition Tn, such that f o r k = 1, 2,

(1.6) [[f - - S(k) llLp(I) ) < C E ( f , I(k))p, j = 0 . . . . . n - 1,

where C depends on max] -1 I/j• I / [ / j l, and also on p as p --+ O.

The proof of Theorem 1.2 is given Section 4. It utilizes the C o piecewise quadratic approximation constructed in Section 2 together with some smoothing technique. The latter uses ideas of Ivanov and Popov [IP] for smoothing piecewise polynomial approx- imants while preserving local approximation errors. Ivanov and Popov used a similar technique to smooth a continuous piecewise quadratic with equidistant knots so that it preserves convexity and stays close to the original function in the sup-norm. The Lp estimates are based on ideas from [HLY].

The most interesting partitions for piecewise polynomial approximation are the uni- form partition and the partitions built on the zeros of the Tchebyshev polynomials. In both cases the ratios [Ij• I/[Ij[ are bounded independently of n. Hence we have absolute constants independent of n, in the above theorems.

Convex Polynomial and Spline Approximation in Lp, 0 < p < Oo 411

2. Convex Piecewise Quadratic Approximation

In this section, we shall consider the Lp [-- 1, 1], 0 < p < c~, approximation of a convex function f by piecewise quadratics. We fix p and fix the function f throughout this section. For any interval I , let E ( I ) := E ( f , I)p be the error in approximating f in the metric Lp( I ) by quadratic polynomials. We say that a quadratic polynomial q is a near best Lp-approximation to f on I with constant Co if

IIf -q l lL , ( l ) < f o E ( l ) .

We shall often make use of the following facts (for proofs of similar results see [DSH, Chapter 3]).

F1. If q is a near best Lp-approximation to f on I with constant Co, i.e.,

IIf - qllL~(l) _< f o E ( I ) ,

then for any interval J with I C J , q is also a near best Lp-approximation to f on J with constant C depending only on IJI / l l I , Co, and p as p ~ 0.

F2. Let I C J be two intervals. If q is a quadratic polynomial satisfying

[If - q l l L y ) <_ f o E ( J ) ,

(note that we do not assume q is a near best Lp-approximation to f on I), then

I I f - qllL~(J) <_ C E ( J )

with the constant C depending only on IJI / l l I , Co, and p as p ~ 0. F3. For any quadratic polynomial q and any interval J ,

IlqllLpCJ> _< !JI1/PlIqlIL~(J) < CIIql[L,r

with C depending only on p as p -+ 0.

L e t Z n := { -1 = : z0 < Zl < --- < zn := 1} be a given partition of [ - 1 , 1 ] and extend this partition by setting zj := - 1 , j < 0, and zj := 1, j > n. We let Jj := [zj, Zj+l], j = 0 . . . . . n - 1. In this section, we shall construct various piecewise quadratic approximations to f which share the convexity of f . We begin by singling out some special points ~j near z j , j = 0 . . . . . n.

1 Let~j := �89 J = 0 . . . . . n - 1 , and 8~ := 7lJn_ll, andlet ~ := [zj, zj +$j ] , j =

0 . . . . . n - 1, Jn := [zn - 6n, z~]. Furthermore, let a~ := [zj-2, zj+3], j = 0, 1 . . . . . n. Throughout this section, we shall use C to denote a constant which depends only on

n - 1 maxj=0 IJj• I/IJj I, and p as p ---> 0. We begin with the following simple lemma:

^

L e m m a 2.1. There are points ~j E Jj, j = 0 . . . . . n, such that the polynomial qj which interpolates f at ~i, i = j - 1, j , j + 1, is a near best Lp-approximation to f on ~ :

IIf -q j l lL , (~ ) < C E ( ] j ) , j = 1 . . . . . n - 1.

Proof. We begin with a best quadratic polynomial approximation pj to f on Jj, j = 0 . . . . . n - 1, and we put p - t := p0, and Pn+l := Pn := Pn-1. Then,

(2.1) IIf--PjllLp(Jj) = E(J j ) , j = 0 . . . . . n - - 1.

412 R.A. DeVore, Y. K. Hu, and D. Leviatan

It follows that

I I f - -Pj - -~IIL~)+II f - -Pj I IL~<~)+II f - -Pj+I l ILp~) ~ C E ( [ z j - l , z j + 2 ] ) , j = 0 . . . . . n.

Hence, for each j = 0 . . . . . n, we can find a point ~j 6 ~ , such that

(2.2) If(~j) - Pi(~j)l <_ C I ~ I - 1 / P E ( [ z j - I , zj+2]), i = j - 1, j, j + 1.

Now let qj be the quadratic polynomial which interpolates f at the points ~i, i = j - 1, j , j + 1. Then, from (2.2), for j = 1 . . . . . n - 1,

Iqj(~i) - Pj(~e)I -< C I ~ I - 1 / p E ( Y j ) , i = j - 1, j , j + 1.

Considering the spacing of the points ~i, we see that

Ilqj - Pjllz~<~ < Cl~ l l /P l lq j -- PjIIL~<~> < C l ~ l 1/p max Iqj(~i) - Pj(~i)l - - - - i = j - l , j , j + l

<_

Hence,

IIf -qjllLp(~) < C(l l f - P j I I L o ( ~ ) + IlPj --QjIILp(~)) < C E ( J j ) . �9

We will build a continuous piecewise quadratic from the polynomials qj. For this, we shall use the following lemma:

L e m m a 2.2. Let - 1 < (o < (1 < �9 �9 �9 < (n < 1 be arbitrary points and let f be a

convex funct ion on [ - 1, 1]. For each j = 1 . . . . . n - 1, let ~j be the quadratic polynomial

which interpolates f at the points (i, i = j - 1, j , j + 1. Then the funct ion

(2.3) ~(x) := /31(x), x 6 [ - 1 , ~'1], max(/3j(x),/3j+l(x)), x 6 [~), ~)+1], /~n_l(X), X C [~n-1, 1].

j = l . . . . . n - 2 ,

is a convex C o ptecewise quadratic with breakpoints at the (j, j = 1 . . . . . n - 1.

Proof. See K. A. Kopotun [KOP] and K. G. Ivanov and B. Popov lIP] where similar constructions have been used. �9

We apply Lemma 2.2 to the polynomials qj and the points {~j} and denote by go the resulting C O piecewise quadratic. We note that on each interval [~j, ~j+l], the function go is identical with one of the polynomials qj or qj+l.

Now go has many of the desired properties but its breakpoints ~j are (near but) not exactly the zj . We are going to modify go to obtain a piecewise quadratic g which has all the desired approximation properties and, in addition, has its break points exactly at the zj . We let rj be the (convex) quadratic which interpolates go at zi, i = j - 1, j , j + 1, j = 1 . . . . . n - 1. We apply Lemma 2.2 to the polynomials rj and the interpolation points zj to obtain a C O piecewise quadratic g with breakpoints at the z j , j = 1 . . . . . n - 1.

Convex Polynomial and Spline Approximation in Lp, 0 < p < oo 413

Theorem 2.3. Let f E Lp[-1, 1], 0 < p <_ cx~, be convex. Then, the continuous convex piecewise quadratic g on the partition Zn defined above satisfies

(2.4) IIf - gllL;(jj) - CE([zj-4, zj+s]), j = 0 . . . . . n - 1.

Proofi On the interval Jj, g is identical with rj or with rj+t. In the case j = 0, this choice is r l and in the case j = n - 1, this choice is r~- l . We shall first estimate

IIf - rj [IL, cJj), t < j < n -- 1. From Lemma 2.1 we have

(2 .5 ) I l f - rjllLp(j:) <_ C ( l l f - q j l l L , @ ) + [Iqj - rjllL,(Sj))

<_ C ( E ( ~ ) + Ilqi -rjllL~(sj)).

For the the sake of convenience in notation in the proof below, we need to set qn : : qn-1 and q_l : = q0 : = ql. Now recall that at each point zi, i = j - 1, j , j + 1, rj interpolates the value gO(Zi) and this value is the same as either qi-l(Zi) or qi(zi). Therefore, from Lemma 2.1,

(2.6) [Iqj -- rjIILAJj) < ClJjl-1/Pl[qj-rjllLo~(Jj) <ClJj[ -1/p max [qj(z i ) -r j (z i ) l i:j--l , j , j+l

<_ CIJjl -lIp m a x I lq~--q~l lL~tzj_ , ,z j+, l j-2<k<j+l

< C max Ilqj - qkllz+,tzj_,,zj+,l j-2<k<j+l

_< C ( l l f - qjllLAzj-,.zj+,l + max IIf--q+llLAzj_,,zj+,l) j-2<_k<j+l

< C(E(J j ) + max E(Jk)) < E([zj-4, zj+4]). j-2<k<j+l

Using this in (2.5) shows that for j -- 1 . . . . . n - 1,

(2 .7) I l f - rj IIz~(J~) <-- CE([zj-4, Zj+4]).

The same analysis with rj replaced by rj+l gives for j = 0 . . . . . n - 2,

[If - r j + l [ILp(Jj) <-- CE([zj_3, Zj+5]).

We have therefore proved (2.4).

3. Construction of Convex Polynomials

In this section, we shall prove Theorem 1.1 on approximation by convex polynomials. We fix the convex function f and 0 < p < c~. To prove Theorem 1.1 we shall first approximate f by a convex C o piecewise quadratic g given by Theorem 2.3 and then approximate g by an algebraic polynomial. This method of proof has been used several times before but we shall follow most closely the constructions and notation in Kopotun [KOP] (although our notation does vary slightly from Kopotun's especially in the ordering of the knots).

Let Xn, j :~--m- cos[(n - j)/n]:rc, j = 0 . . . . . n, Xn,j := - - l , j < 0, and xn,j := 1, j > n. We will apply the results of Section 2 with the zj = xn,j for all j . We let J,,j , ]~,j, etc., be the obvious analogues of the intervals in that section. Note that the ratios [J,,j+l [/I Jj]

414 R.A. DeVore, Y. K. Hu, and D. Leviatan

are bounded by a constant independent of n and j , so that the estimates of Section 2 hold with absolute constants (except for the dependence on p).

We shall use some results of Kopotun [KOP] on the approximation of truncated powers. To describe these, we let

I J , , j l Xn,j := Xtx,,j,1] a n d l~rn'J : = IX -- Xn,jI + IJ , , j I ' 1 < j < n - 1 .

Then Kopotun [KOP, Lemma 2] has constructed three sets of polynomials an,j, Rn, j ,

and Rn , j , j = 1 . . . . . n - 1, of degrees not exceeding some fixed multiple of n, which approximate the truncated powers. Kopotun's applications were primarily in the L ~ - norm. However, we shall need estimates in the Lp-nOrm, and as p ~ 0, we need better and better estimates. The following estimates can be obtained from Kopotun's proof on a closer look. We shall not indicate the dependence of the polynomials on p and all constants C depend at most on p as p ~ 0. The polynomials an,j, Rn, j , and Rn, j ,

j = 1 . . . . . n - 1, are of degree not exceeding max{ 1, 1/p}50n and .satisfy the following estimates:

_ ('dtmax{l'l/p}1611 .I I(x --Xn,j)+ - -an , j ( x ) [ < ~wn , j .~n,j., X E [--1, 1], (3.1)

(3.2)

and

(3.3)

_ ("'drmax{l'l/p}1611 .i 2 [(X X n , j ) 2 - - e n , j ( x ) [ < ~ ' r n , j ,~n,jt , x ~ [ - 1 , 11,

X 2 _ (,drmax{1,1/p}1611 .i 2 I( x - - n , J ) + - g n , j ( x ) l < ~ ' n , j ,~n , j , , X ~ [ - - 1 , 1].

Also, it is possible to prescribe an (sufficiently large) integer M so that for all n and all j = l . . . . . n - l ,

(Xn,j+ 1 -- Xn , j )a~n ,M j (X) -- R" Mn,Mj(X) 7> - -2Xn, j (x) , x e [ - 1 , 1], (3.4)

and

( 3 . 5 ) tl --H

(Xn,j -- Xn,j-1)GMn,M j (X) + RMn,M j (X) > 2X. , j (x), x ~ [ - 1 , 1].

Note that

(3.6) [J.,j[ ~ I Jg . , g j [ ,

with constants in this equivalence that are independent of n and j = 0 . . . . . n - 1.

X . r t Now let g . be the function of Theorem 2.3 for f and the points { n,j}j=0. We can

represent gn as a sum of the truncated powers (x - xn, j) + and (x - xn, j) 2, j = 1 . . . . . n. As in Kopotun [KOP], we shall classify the knots x . , j according to four types depending on the second-order divided differences of gn:

(3.7) an,j := [xn,j-1, xn,j , xn, j+l]g, , j = 1 . . . . . n - 1.

For the following comparisons, we define a,,o = a,,~ := oc. Let 1 < j < n - 1. We define Xn,j to be of type I, if

(3.8) a~,j+i < an,j < an,j-1.

We define x,,,j to be of type II, if

(3.9) an, j -1 < an,j <__ an, j+l ,

Convex Polynomial and Spline Approximation in Lp. 0 < p < oo 415

We define Xn,j to be of type III, if

(3.10) max{an, j_ l , an,j+l} < an,j,

All other Xn,j'S are defined to be of type IV. Note that Xn, 1 can only be of type I or type IV, and Xn: - I can only be of type II or type IV.

In order to represent gn as a sum of truncated powers we define

An, j : = an,j - an,j+ l, j = 1 . . . . . n -- 2, Bn,j : = - - A n , j - l , j = 2 . . . . . n - 1,

and

An,o := [xn,o, Xn.1]g - [Xn,1, Xn,2]g -t- [Xn,O, xn,2]g.

It follows from the definitions of type that An, j > 0 for Xn,j of types I or III and B~,j > 0 if xn,j is of types II or III.

Using the divided difference representation of piecewise polynomials, we obtain (see [KOP]) the following representation for gn for x 6 [--1, 1]:

(3.11) gn(X) = g ( - 1 ) + An,o(x + 1) +an, l(X + 1) 2

+ ~ a n , j ( ( X n , j + l - X n , j ) ( x - x . , j ) + - ( X - X n , j ) 2 + ) xn,j E I U nI

+ ~_. Bn, j ( (x . , j - xn, j -1)(x - Xn,j)+ + (X -- X.,j)2+). x.,j 6 II tO nI

We shall next estimate An , j . We fix n and let g = gn and zj := Xn,j for all j . Let pj be a best L p [ Z j - l , Z j+2] approximation to f by quadratic polynomials. We recall, that on each interval Jj := [zj, Zj+l], g is either rj or rj+l with the rj the polynomials of Section 2. Hence,

IAn,jl = ( z j + 2 - z j -1) I [ z j - 1 , zj , Z j + l , zj+2](g - Pj)I

< C I J j l - 2 1 1 g - PjllL~tzj-,,z:+2] < CIJj1-2 max Ilri -- PjllL~[zi-,,zj+2] j - l<i<j+2

< CIJj1-2-alp max [Iri - pjllL:zj_,,zj+21. j - l<i<j+2

We write r i -- p j -m- f - p j - ( f - ri) and use (2.7) to obtain

IAn.jl <- C t J j I - z - I / p E ( [ z j - 5 , z j + 6 ] ) ,

where the constant C depends at most on p as p ~ 0. With the notation J.* " - j,n "-- [Xn,j-6, Xn,j+6] , w e o b t a i n

(3.12) IAn,j[, Inn,jl _< CIJ j l -2 - t /pE(J~ ,n ) .

AS our approximation to f , we take the polynomial

Pn(x) := g ( - 1 ) + An,O(X + 1) +an , l (X + 1) 2

+ Z an, j ((Xn,j+l -- Xn,j)O'Mn,Mj(X ) -- RMn,Mj(X)) x. , jEIUIII

+ ~_. B , , , : ( ( x . . j - x n , j _ ~ ) ~ n , . j ( x ) + - ~ . n , . j ( x ) ) . x,,,j E n u In

416 R.A. DeVore, Y. K. Hu, and D. Leviatan

Note that P, is a polynomial of degree at most 50Mn max(l , I/p) and that we keep the An, j'sand Bn, j 'S and the knots x,,,j at level n while taking the polynomials ~r, R, and R at the level Mn. However, one should observe that xn,j = XM,,,Mj for all j = 0 . . . . . n, Now by virtue of (3.4) and (3.5), it follows that

P~(x)>" _ g ' / ( x ) _ > 0 for all x c [ - 1 , 1 ] , x#xn , j , l < i < n - 1 .

Thus Pn is convex on [ - 1 , 1]. We next estimate Ilgn - enlltA-l,lJ in the case 1 < p < c~. We claim

(3.13),lgn Pn p - - f l j~i u - [[Lp[-1,1] - - An,j[(Xn,j+l--Xn,j)((X--Xn,j)+--ffMn,Mj(X)) 1 xn. III

-- ((X --Xn,j)2+ -- R M n , M j ( X ) ) ]

+ Z n n ' j [ ( X n ' j -- X n ' j - l ) ( ( x -- Xn ' j )+ -- (7Mn'Mj(X)) x.,j EIItAIII

-~M,, ,Mj(X))] P + ( ( x - x. , i)2+ - d x

f <_ C IAn,jl 2 16 [JMn,Mj] ~M,,,Mj(X) 1 xn. III

2 16 ]P + ~ IB~,jlIJMn,MjI 7rM,,,Mj(X) dx

x/,,j EIIUIlI

n--1 fl n--I Z 16 * p _ < C [ J n , j l - l E ( J ~ , j ) p ~ M n , M j ( X ) d x < _ C E ( J n , j ) . j = l --1 j = l

Indeed, the first inequality in (3.13) uses (3.1), (3.2), and (3.3), the second inequality uses Jensen's inequality and our estimates (3.12) for the An, j and the Bn,], and the last inequality uses the straightforward estimate

f l ~ 26 dx < CIJ~,jl. 1

The estimate (3.13) also holds for 0 < p < 1 with the only change in the proof is that one uses the subadditivity of II �9 IIP~t_l lj.

Analogous to (3.13), we obtain the estimate

n-1 P (3.14) Ilf - g, [ILA-I,1 ] <- C ~ E(J*,j) p,

j=l

by adding the local estimates (2.4). Therefore, we have

n-I e P * p (3.15) IIf -- nllL~[-~,lj <-- C y ~ E(Jn,j) ,

j=l

Convex Polynomial and Spline Approximation in Lp, 0 < p < cx~ 417

By Whitney's theorem (1.5), we have

(3.16) E ( J ) <_ Cco3(f, lJI; J)p

for each interval J and with a constant depending only on p as p ~ 0. Also, it is proved in [DLY] (see, e.g., (4.1) and (4.5) there) that for any interval J 6 [ -1 , 1]

f j f IJlA~ qg(x) 3 (3.17) o~3(f, I/I; J)pP -< C a0 IJI ]Ah~(x)(f 'x; J)]P d h d x

fC2/n f j <_ Cnjo 3 IAh~o(x)(f, X; J)l p dx dh.

Note that in [DLY], (3.17) is only stated for 0 < p < 1 but the proof is the same for all p. Using (3.16) and (3.17) in (3.15), we obtain

[if p CnfoC2/"f 1 -- P, zIIL,[-I,1] < [A3~(x)(f, x)[ e d x d h < Co9y(f, 1/n)p p, 1

where we have used the fact that any x 6 [ -1 , 1] appears in at most 12 of the intervals Jj*,n. We have therefore proved Theorem 1.1. �9

4. Smoothing Lemmas and the Proof of Theorem 1.2

In Theorem 2.3 of Section 2, we have constructed a convex C O piecewise quadratic with good approximation properties. In this section, we shall use this piecewise quadratic and certain methods of smoothing to prove Theorem 1.2.

The following two lemmas will be used to eliminate discontinuities of piecewise polynomials:

Lemma 4.1. Let a < 0 < b, and let h(x) := 13x+ with ~ > O. Then there exists a unique C 1 quadratic spline s on [a, b] such that the only breakpoint o f s in (a, b) is a simple knot at 0 and

(4.1) s(J)(a) = h(J)(a), s(J)(b) = h(J)(b), j = 0, 1,

and

(4.2) s ( J ) ( x )>O, x 6 [ a , b ] , x # 0 , j = 0 , 1 , 2 .

Moreover,

(4.3) IIs - hllLA.,b ] < C IlhllL,,[a,b3 < C~ blq-1/p, where C depends only on the ratio lal/b, and on p if p ---> O.

Proof. Any C l quadratic spline on [a, b], with the only knot at 0, can be represented in the form

s(x) = ko + kl (x - a) + k2(x - a) 2 -q- k3x 2.

Simple computations show that the only such spline which also satisfies (4.1), is given by

fib ( x - a ) 2 + f l ( a + b ) 2 s(x) -- 2a(b - a) -~-~ x+.

418 R.A. DeVote, Y. K. Hu; and D. Leviatan

NOW,

s ' ( x ) = a( a ) ' x < O,

b ( b - a ) ' x > 0 ,

whence s" > 0 on [a, b]. Also, s ' is piecewise linear (with breakpoint at 0) such that s ' (a) = O, s'(O) : 13b/(b - a) > O, and s '(b) = h'(b) : / 3 > 0. Thus s ' > 0 on [a, b] and s o s is nondecreasing. Since s(a) = 0, we have s > 0 on [a, b] and the proof of

(4.2) is complete. In order to prove (4.3) we write

IlsllLpta,bl <-- (b - a)l/Pllsllcta,bl = (b - a ) l / p s ( b )

= flb(b - a ) t/p < C f l b I+I/p -m- CIIhllL~I~.bl,

and (4.3) readily follows. �9

L e m m a 4.2. Let a < 0 < b, zl = 0, 0 < Z2 < b, and let h ( x ) := yx~_ with y > O. Then there exists a unique C 2 cubic spline s on [a, b] such that the only breakpoints o f

s in (a, b) are two simple knots zl and z2 and

(4.4) s (j) (a) = h (j) (a),

Moreover, we have

(4.5)

and

(4.6)

s(J)(b) = h(J)(b), j = O, 1, 2.

s~>(x) >_ O, x c [a, b], j = O, 1, 2,

IIs - hllL,ta,b] ~ C IlhllLpEa,b] ~ C y b2+l/p,

where C depends on the ratio [al/b, and also on p i f p ~ O.

Proof. As in the proof of the previous lemma, straightforward computations show that the only C 2 cubic spline on [a, b], with knots at Zl = 0 and z2, 0 < z2 < b, which satisfies (4.4), is

ybz2 (x - a) 3 + y (ab + az2 + bz2)x3 + s (x ) -~ 3a(b - a)(z2 - a) 3abz2

Fab + 3z2(z2 - a) (b - z2) (x - z2) 3.

We observe that s" is piecewise linear (with breakpoints at 0 and at z2) such that s" (a) = 0,

2Fbz 2 2F(b 2 - ab - az2) s"(O) = (b - a)(za - a) >- O, s"(z2) -- b(b - a) > 0

and s ' ( b ) = h"(b) = 2y _> 0. Thus s" >_ 0 on [a, b], which together with s ' (a) = O,

implies that s ' > 0 on [a, b]. Finally, the latter implies that s is nondecreasing and together with s(a) = 0 we get that s > 0 on [a, b]. This completes the proof of (4.5).

In order to prove (4.6) we write

llsqqLAa,bl < (b -- a ) 1/p l[sl[c[a,b] = (b - a) l /Ps(b)

= F b 2 ( b - a) lip <_ C y b 2+l/p = CIIhllcpta,bj.

This evidently yields (4.6).

Convex Polynomial and Spline Approximation in Lp, 0 < p < oo 419

The following theorem will prove the case of Theorem 1.2 dealing with C 1 piecewise quadratics. We use the notation of the introduction and Section 2. Given the partition

t" n T n = { J} j=0 ' w e r eca l l o u r n o t a t i o n I~ 1) : = [ t j - l l , t j + l l ] .

T h e o r e m 4.3. For each part i t ion Tn, there is a convex, C 1 p iecewise quadratic G on Tn that satisfies

(4.7) IIf - G[IL//j) < CE(Ir j = 0 . . . . . n - 1,

where C depends on max0_<j<~ (l I j+l I / l l j l) and on p as p --+ O.

Proof. We define zj := tzj, j = 0 . . . . m, with m := [(n + 1)/2] and let Zn := {zj }~n= 0, and we let the intervals Ij := [tj, tj+l] correspond to the partition Tn and the intervals Jj := [zj, zj+l] correspond to the partition Zn. Let g be the convex, C ~ piecewise quadratic of Theorem 2.3 for the partition Zn. We shall use Lemma 4.1 to add a knot t2 j+l o n each interval [z j , z j+l] and to create a C 1 piecewise quadratic G which has the desired properties. On each interval Jj, g ---- gj with gj either rj or rj+l ; the quadratic polynomials rj are defined in Section 2 preceding Theorem 2.3.

We consider g on any interval Kj := [t2j-1, tzj+l]. Since g is continuous with only one breakpoint (at tz j) on this interval, we have

g ( x ) = g j _ l ( x ) + o t j ( x - z j ) Z + f l j ( x - z j ) + , x c K j .

Since g is convex, we have flj > O. We apply Lemma 4.1 to obtain a C 1 piecewise quadratic sj on Kj with a simple knot at zj that approximates f l j (x - z j )+ as described in that lemma. We then define

G ( x ) = g j _ l ( X ) + ~ j ( x - z j ) 2 + s j ( x ) , x E Kj , j = 1 . . . . . n - 1

and G ( x ) = g (x ) for all other x E [ - 1, 1]. Because of (4.1), G is in C l and is a piecewise quadratic on the partition Tn. Because gj-1 (x) + ~j (x - zj) 2 has a nonnegative second derivative, property (4.2) guarantees that G is convex on each Kj. Because G is in C 1, it is convex on [ - 1 , 1]~

To complete the proof of the theorem, we need only show that

(4.8) IIg - G I I c , o:~ <- C E ( [ z j - 5 , zj+s]).

According to (4.3), we have

(4.9) IIg - GIIL~(Kj) = II/~j("--zj)+ --sjlILp(K D < C f i j l h j l 1§

We now estimate flj in order to prove (4.8). We have

]flj[ = [g~j(zj) -- g j_ l ( z j ) l < llg~ - gj-lllL~(Jj_,)

<_ CIz) - z j -a l -a l lg j - gj-lllL~(Jj_~)

<-- ClJj-ll• - gJ-lllLrCJj 1)

<_ CIJj-11-(I+I/p) E ( [ z j -5 , zj+s]),

where the last inequality is derived as in (2.4). Using this in (4.9) gives (4.8). �9

420 R.A. DeVore, Y. K. Hu, and D. Leviatan

Remark. We have already mentioned in the Introduction that Theorem 4.3 for a partition Tn of equidistant knots and approximation in the max-norm was proved by Ivanov and Popov [IP]. They inserted two additional knots between any two knots of the piecewise quadratic g. That it suffices to insert one knot between any two knots of g, was first pointed out to us by Dietrich Braess.

Finally, we consider the case of approximation by C 2 piecewise cubics. For the par- tition Tn : = (tj), we let 1) 2) := [tj_49, tj+49 ].

The o rem 4.4, For any partition Tn, there is a convex C 2 cubic spline s on Tn, such

that,

(4.10) [ I f - sllL~ij~ <_ CE(IS2)), j = 0 . . . . . n - 1,

where C depends on maxo<j<_n(llj• and on p as p --+ O.

Proof. We define Zn : = {zj }jm 0 where zj := t4j, j = 0 . . . . . m, and m : = [(n + 3)/41.

According to Theorem 4.3 (for the partition Zn), there is a convex, C 1 piecewise quadratic G defined on Zn, which satisfies the conditions of that theorem. We will use Lemma 4.2 to smooth G at the interior knots z j , j = 1 . . . . . m - 1. We let G = : Gj on Jj := [z j , z j+l ]. Then, each Gj is a convex quadratic polynomial and we have

G j ( x ) - G j _ l ( x ) = Vj(x - z j ) 2.

If yj > 0, then for Kj : = [t4j-2, t4j+2] and j = 1 . . . . . m - 1, we have

G(x) = Gj_I (X) -I- y j ( x - z j ) 2, x c Kj .

In this case, we let sj be the C 2 piecewise cubic with simple knots at t4j, t4j+l given by Lemma 4.2 for the function h j ( x ) = y j (x - z j ) 2 and a = t4j-2, b = t4j+2. We define

s (x ) = Gj_ l (X) + s j (x ) , x c Kj .

Note that sj (x) has its simple knots at t4j, t4j+l and matches yj (x - z j)2+ in value and first and second derivatives at the points t4j-2 and t4j+2. Note also that in the case j = m - 1, the points t4j+l and t4j+2 may both equal one. In this case, we set Sm-1 : = 0.

If yj < 0, we have

G(x) = G j ( x ) - y j ( z j - x)~_, x c Kj .

In this case, we let sj be the C 2 piecewise cubic of Lemma 4.2 with the simple knots at zj and 2zj - t4j-1 for the function h i (x ) = - g j ( x - z j ) 2 and a = 2zj - t4j+2,

b = 2zj - t4j-2. We define

s (x ) : = G j ( x ) + s j (2z j - x), x 6 Kj .

Note that sj (2zj - x ) has its simple knots at t4j-1, t4j and matches - y j (zj - x)2+ in value and first and second derivatives at the points t4j-2 and t4j+2.

Because of (4.4), s is a C 2 piecewise cubic spline on the partition Tn. Also, sj is convex because of (4.5). Hence, s is also convex on [ - 1 , 1]. On the interval Kj , we have

Convex Polynomial and Spline Approximation in Lp, 0 < p < ~ 42i

from (4.7) and (4.6),

(4.11) l l f - SI[Lp(Kj) ~ I I f -- GIIL~(Kj) + [IG - slIL~<K:)

< CE([zj-12, Zj+I1]) q- ClYjlltaj+2 - t4j-212+l/p.

To complete the proof, we need to estimate I Yj I. We have

I I I I (4.12) 21y:1 = IGj ( z j ) - Gj_ 1 (zj)t

= flay - aj_111L~<::) <_ CIJjI-ZI[Gj - G:-III/~<Jj) <_ CIJjI-Z-I/OIIGj - Gj-~IIL~(J,).

We write Gj - Gj-I = (f - Gj-O - (f - Gj) to find

IIGj - Gj-lllp <_ C( l l f - Gj[IL,(j:) + I l f -- Gj-IlILp(J:))

< CE([ z : -n , z:+ll]) + E([zi-12, z/+10]),

where we have used (4.7) and F2. If we use this last inequality in (4.12) and then in (4.11), we obtain

[ I f -- SItLs<K :) < CE([z j - l z , zj+l~]).

Recalling that zj = t4j and Kj = [t4j-2, t4j+2], we see that we have proved (4.10). �9

Acknowledgment . The authors thank the referee for suggesting a significant improve- ment of the proof of Lemma 4.2, which made it both shorter and simpler than the authors' original proof. The first and third authors were supported in part by the BSF Grant 89- 00505. The second author was partially supported by the Faculty Research Committee of Georgia Southern University.

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R. A. DEVORE, D. LEVIATAN (1993): Convex polynomial approximation in Lp (0 < p < 1). J. Approx. Theory, 75:79-84. R. A. DEVORE, D. LEVIATAN, X. M. YU (1992): Polynomial approximation in Lp (0 < p < 1). Constr. Approx., 8:187-201. R. A. DEVORE, G. G. LORENTZ (1993 ): Constructive Approximation. Grundlehren Series, Vol. 303, New York: Springer-Vedag. R. A. DEVORE, R. C. SHARPLEY (1984): Maximal functions measuring smoothness. Mere. Amer. Math. Soc., 293:1-115. Z. DITZIAN, V. TOTIK (1987): Moduli of Smoothness. New York: Springer-Verlag. Y. K. Hu, D. LEVlATAN, X. M. Yu (1994 ): Convex polynomial and spline approximation in C [-1, 1]. Constr. Approx., 10:31-64. K. G. IVANOV, B. PoPov (1996): On convex approximation by quadratic splines. J. Approx. Theory, 85:110-114. K. A. KOPOTUN (1994): Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials. Constr. Approx., 10:153-178. D. LEVIATAN, X. M. Yu (1986): Shape preserving approximation in LP. Preprint.

422 R.A. DeVote, Y. K. Hu, and D. Leviatan

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A. S. SVEDOV ( 1981): Orders ofcoapproximation offunctions by algebraicpolynomials. Translation from Mat. Zametki, 29:117-130. Math. Notes, 29:63-70. X. M. YU (1987): Convex polynomial approximation in Lp spaces. Approx. Theory Appl., 3:72-83.

R. A. DeVore Department of Mathematics University of South Carolina Columbia SC 29208 USA

Y. K. Hu Department of Mathematics and Computer Science Georgia Southern University Statesboro GA 30460-8093 USA

D. Leviatan Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv 69978 Israel