on some aspects of multivariate polynomial interpolation

Advances in Computational Mathematics 12 (2000) 311–333 311

On some aspects of multivariate polynomial interpolation

A. Le MehauteDepartement de Mathematiques, Universite de Nantes, Faculte des Sciences et des Techniques,

2 rue de la Houssiniere, B.P. 92208, F-44322 Nantes Cedex 03, FranceE-mail: alm@math.univ-nantes.fr

The purpose of this paper is to present some aspects of multivariate Hermite polynomialinterpolation. We do not focus on algebraic considerations, combinatoric and geometricaspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolationand some higher dimensional problems. The concepts of similar and equivalent interpolationschemes are introduced and some differential aspects related to them are also investigated.

Keywords: Hermite interpolation, multivariate interpolation, Lagrange formula, Newtonformula, Aitken-type algorithm, Abel interpolation

1. Introduction and notations

The aim of this paper is to review some known results, but also to presentsome work under investigation, and perhaps to raise some interest in these problems,which have the attention of people from various fields of pure and applied mathematics,including number theory, algebraic real geometry, real analysis and of course numericalanalysis and approximation theory.

In this section we recall some results of classical real algebraic geometry relatedto interpolation. In section 2 we extend and complement some of the aspects ofmultivariate Hermite polynomial interpolation considered in [19]. For error boundswe refer to [15–17]. Finally, in section 3 we give some remarks on multivariateinterpolation formulas for Lagrange, Hermite and Abel interpolation.

Let Ω ⊂ Rd be a bounded open domain and PN = PN [Rd] the space of d-variatepolynomials whose dimension is U(N , d) =

Definition 1.1. An interpolation scheme is a triplet (A,PN ,L), where A is a set ofnodes (data points) in Ω ⊂ Rd,

A =ari

06r6s;16i6Nr ,

L =ϕ0i

16i6N0

∪ϕrij

16r6s;16i6Nr ;16j6diris a set of N linearly independent linear forms with supports in A defined by order ofderivation and vectors (directions) of differentiation.

J.C. Baltzer AG, Science Publishers

312 A. Le Mehaute / Multivariate polynomial interpolation

More precisely, for any v ∈ Cs(Ω), with s an integer sufficiently large,

ϕ0i : v 7→ ϕ0

i [v] = v(a0i

), 1 6 i 6 N0,

ϕrij : v 7→ ϕrij[v] = Drv(arij)· ξrij , 1 6 r 6 s; 1 6 i 6 Nr; 1 6 j 6 dij .

Obviously, according to the above notation one has

N = N0 +s∑r=1

Nr∑j=1

which is exactly the dimension of the polynomial space PN .The interpolation scheme is said to be PN [Rd]-unisolvent, or, briefly, unisolvent,

if and only if there exists a basis B orthonormal in the duality between L and PN , i.e.,

B =p0i

16i6N0

∪prij

16r6s;16i6Nr;16j6dir

is a basis of PN [Rd] such thatϕ0i [p

0i ] = δij , 1 6 i 6 N0; 1 6 j 6 N0,

ϕ0i [p

rik] = 0, 1 6 i 6 N0; 1 6 r 6 s; 1 6 j 6 Nr; 1 6 k 6 dir ,

ϕrij[p0k] = 0, 1 6 r 6 s; 1 6 i 6 Nr; 1 6 j 6 dir; 1 6 k 6 N0,

ϕrij[ptk`] = δrtδikδj`,

1 6 r, t 6 s; 1 6 i 6 Nr; 1 6 k 6 Nt;1 6 j 6 dir; 1 6 ` 6 dkt.

Let us be given a set of U(N , d) real numbers

Y =y0i

16i6N0

∪yrij

16r6s;16i6Nr ;16j6dir .

In case of unisolvency of (A,PN ,L), the interpolation problemFind a polynomial HA,L ∈ PN such that

HA,L(a0i ) = y0

i , for 1 6 i 6 N0,

HA,L(arij) = yrij , for 1 6 r 6 s; 1 6 i 6 Nr; 1 6 j 6 dir

has one and only one solution.Using the dual basis, the interpolant HA,L is easily seen to be defined by

HA,L[v] =

N0∑i=1

ϕ0i [v]p0

i +s∑r=1

Nr∑j=1

dir∑j=1

ϕrij[v]prij ,

which reduces for Lagrange interpolation to

LA[v] =N∑i=1

ϕi[v]p0i .

A. Le Mehaute / Multivariate polynomial interpolation 313

Remark. One of the main difficulties when dealing with multivariate polynomial in-terpolation remains the inability to formulate such bases. It will be the topic of ourthird section but unfortunately for some specific cases only.

Let D(A) be the determinant associated to the linear system which defines HA,Labove. We have the following:

Definition 1.2 [21]. The interpolation problem is said to be

(i) regular if D(A) vanishes for no choice of knot set A,

(ii) singular if D(A) = 0 for all knot set A,

(iii) almost regular if D(A) vanishes only on a subset of Rd of measure zero.

Almost regularity of a problem does not imply unisolvency for a given set A.We need a characterization of the sets which produce unisolvency. In the plane wecan use the following results from algebraic geometry.

For a data point A ∈ R2, let us denote by δA the pointwise evaluation operator,i.e., δA[f ] = f (A) and more generally, δαA for the pointwise evaluation operator of thederivative of order α, i.e., δαA[f ] = ∂αf (A).

Definition 1.3. A point A ∈ A = SuppL has multiplicity (p + 1) for (A,PN ,L) ifδαA ∈ spanL, for all α such that |α| 6 p and there exists one β such that |β| = p+ 1and δβA /∈ spanL.

Using the canonical basis for PN and evaluating the determinant, we obtain:

Proposition 1.1. The interpolation problem has no solution for general Y if and onlyif there exists an algebraic curve of degree N that passes through all the Ai ∈ A, eachof them being counted with its multiplicity mi.

Corollary 1.1. An interpolation scheme (A,PN ,L) is unisolvent if and only if theonly polynomial of degree N that vanishes on A, with the multiplicities, is the zeropolynomial.

This provides a very efficient tool when it is easy to describe the curve, forexample if one can decompose the curve as a product of elementary ones. For this,it is convenient to use the following result, known as Bezout’s theorem in algebraicgeometry [28]:

Theorem 1.1. Two curves of degree m and n respectively, with no common com-ponent, intersect each other at exactly mn points (real or complex), not necessarilydistinct.

From this we derive the following corollary [17,19]:

Corollary 1.2 (Decomposition theorem). Let f be a polynomial of degree d, with mzeros Ai, each of them with multiplicity mi. If M of the Ai, say A1,A2, . . . ,AM , lieon the same straight line ` (whose equation is ` = 0), and if

∑Mi=1 mi > d, then there

exists a polynomial f1, of degree d− 1, such that f = `f1. Moreover, if Ai is locatedon `, its multiplicity for f1 is mi − 1, and if Ai is not on `, its multiplicity for f1 isstill mi.

In Rd, for d > 2, Bezout’s theorem does not apply, at least under the form wehave used, but fortunately, it is possible to prove a decomposition theorem:

Proposition 1.2. Let P be a polynomial of degree k in Rd, which is identically zeroon an affine hyperplane H whose equation is `(x) = 0, ` ∈ P1[Rd]; then there existsa polynomial Q ∈ Pk−1[Rd] such that P = Q`.

The results are not so easy to work with; there are still some sufficient conditionsfor the characterization of the unisolvency of interpolation schemes, the simplest onebeing:

Proposition 1.3. Let (A,Pk[Rd],L) be an interpolation scheme in Rd, with cardL =U(k, d). Let us assume there exists a hyperplane H such that cardL|H = U(k, d− 1).Then (

A,Pk[Rd],L)

is Pk[Rd]-unisolvent

if and only if the two following properties are fulfilled:(A ∩H ,Pk

[Rd−1],L|H) is Pk[Rd−1]-unisolvent and(

A \ (A ∩H),Pk−1[Rd],L \ L|H

)is Pk−1

[Rd]-unisolvent.

This leads to:

Proposition 1.4. Let V be a space of polynomials and m = dimV . Let L ⊂ V ′

with cardL = m, and assume that there exist k hyperplanes Hi, i = 1, . . . , k whoseequations are `i(x) = 0 such that

(1) L =⋃ ki=0Li, with suppLi = Ai ⊂ Hi, i = 1, . . . , k,

(2) (Ai,V|Hi ,Li) is V|Hi-unisolvent and (A0,V ∩W ,L0) is V ∩W -unisolvent, with

W = q(∏ki=1 li); q polynomial.

Then (A,V ,L) is V -unisolvent, for A =⋃ ki=0Ai.

2. Differential properties of unisolvent Hermite interpolation schemes

2.1. Taylor interpolation as a limit case

Let us first recall what happens in the univariate case. Let f ∈ Ck+1[a, b] andA = x0,x1, . . . ,xk a set of k + 1 distinct points in [a, b]. Let HA be the Lagrangeinterpolation operator defined on A, i.e., HA[f ] is the only polynomial of degree ksuch that

HA[f ](xi) = f (xi), ∀i = 0, . . . , k.

It is well known that

limxi→a,∀i=0,...,k

HA[f ] = T ka [f ],

where T ka [f ] is the Taylor polynomial expansion of f around a.Consider now the multidimensional case, i.e., d > 1.Let (A,PN [Rd],L) be a unisolvent Hermite interpolation scheme in Rd, with

L = `1, `2, . . . , `N , and let p1, p2, . . . , pN be a basis in PN [Rd].Obviously, (A,PN [Rd],L) is unisolvent if and only if the Gram determinant,

G =∣∣`i(pj)∣∣, i, j = 1, . . . ,N ,

is nonzero.Moreover, from Cramer’s rule, the unique solution HA,L[f ] of the interpolation

problem associated with `i(HA,L[f ]) = fi, i = 1, . . . ,N , is explicitly given as

HA,L[f ] = − 1G

∣∣∣∣∣∣∣∣∣∣∣

0 p1 p2 . . . pNf1 `1(p1) `1(p2) . . . `1(pN )f2 `2(p1) `2(p2) . . . `2(pN )...

......

fN `N (p1) `N (p2) . . . `N (pN )

∣∣∣∣∣∣∣∣∣∣∣,

that is,

HA,L[f ](M ) =N∑i=1

`i[f ]QiA,L(M )

with `i[f ] = fi and QiA,L a polynomial of degree N which is independent of f .Let us assume that A is in a compact domain C in Rd, with

d = diam C 6 1

(for d > 1 we would have some constants, powers of d, appearing in most of theforthcoming equalities or inequalities).

For f ∈ Cm(C), we consider the norm

‖f‖kC = max|α|6k

maxM∈C

∣∣∂αf (M )∣∣.

The operator HA,L : f 7→ HA,L[f ] is a linear operator on Cm(C) which is a projectoron PN [C] with a norm

‖HA,L‖C = maxM∈C

N∑i=1

∣∣QiA,L(M )∣∣.

For the proofs of the following proposition and theorems we refer to [19].

Proposition 2.1. The norm ‖HA,L‖C of HA,L is invariant under a one-to-one affinemapping.

Let C be a convex compact domain in Rd with a non-empty interior subset. Letρ(M ) be the maximum of the diameters of all the spheres inscribed in the simplicesS such that M ∈ S and the vertices of S are on the boundary of C and, finally, let

r = r(C) =12

infM∈C

ρ(M ).

Multivariate Markov inequality [4]. For a polynomial P of degree N and for anyα, we have ∥∥∂αP∥∥C 6

)|α|N2(N − 1)2 · · · (N − |α|+ 1)2‖P‖C.

As seen in [19], this inequality is basic for the proofs of the next theorems.Let us denote

• φ(N , |α|) = N2(N − 1)2 · · · (N − |α|+ 1)2,

• E(f ,m,N , C) = infP∈PN ‖f − P‖mC , and

• ω(f ,m, δ) = max|α|=m maxM1M26δ |∂αf (M1)− ∂αf (M2)|.Then we have:

Theorem 2.1 [19]. Let A,L and C be fixed. Then for m 6 N ,∥∥f −HA,L[f ]∥∥m

C 6 E(f ,m,N , C)

)mφ(N ,m)‖HA,L‖C

Consider now a family of unisolvent interpolating schemes (Ai,Li) for i =1, 2, . . . defined in the following way:

(i) Let (A0,L0) and a point A be given, together with a compact set C0 such thatA0 ⊂ C0 and A ∈ C0.

(ii) Let θi be a sequence of one-to-one affine mappings and let Ci = θi(C0), Ai =

θi(A0) and Li = θi(L0).

Definition 2.1. The family Ai, i = 1, 2, . . . converges regularly to A as i → ∞ ifthe three following assumptions are fulfilled:

(i) A ∈ Ci, i = 1, 2, . . .;

(ii) there exists a (unique) constant H1 such that diam Ci = di 6 H1ri, i.e., di/ri 6H1 uniformly relatively to i;

(iii) limi→∞ di = 0.

(As above, in order to avoid constants in the inequalities, we will assume that di 6 1.)

Theorem 2.2 [19]. Let f ∈ Cm(C0) and assume that the family Ai, i = 1, 2, . . .converges regularly to A as i→ ∞. Then there exists a constant H such that for allk 6 m and α with |α| = k,∣∣∂αHAi,Li[f ](A)− ∂αf (A)

∣∣ 6 Hrm−|α|i ω(f ,m, ri).

Now we denote by TNA [f ] the Taylor polynomial expansion of order N of faround A.

Theorem 2.3 [19]. Let f ∈ CN (C0) and assume that the family Ai, i = 1, 2, . . .converges regularly to A as i→∞. Then there exists a constant K such that for allM ∈ Ci, ∣∣∂αHAi,Li[f ](M )− ∂αTNA [f ](M )

∣∣ 6 KdN−|α|i ω(f ,N , di).

2.2. Comparison between unisolvent schemes

First we shall introduce the concept of similar interpolation schemes.

Definition 2.2. Two interpolation schemes (A,PN ,L) and (B,PN ,L′), both PN -unisolvent, are similar if the only differences are in the location of the nodes andthe direction of the vectors ξrij .

This definition provides a relationship between schemes of the same kind. Givenany scheme related to L we will denote by ΩL the (equivalence) class of all theschemes similar to L.

In order to simplify the notations, whenever (A,L) ∼ (B,L′) we will write only(A) and (B), as well as HA for HA,L or HB for HB,L′ .

Our goal is to “measure” the error made when replacing A by B when dealingwith say Lagrange or Hermite interpolation, thus to have an inequality∥∥HA[f ]−HB[f ]

∥∥rC 6 Kd(A,B)‖f‖rC,

where d(A,B) measures the distance between A and B.We use ΣN for the set of all permutations of N objects.

Definition 2.3. Let A and B two similar schemes, both PN -unisolvent. They are saidto be equivalent, and we write A ≡ B, if there exist permutations σ0,σr,σir such that

σ0 ∈ ΣN0 , σr ∈ ΣNr , σir ∈ Σdir , 1 6 r 6 s, 1 6 i 6 Nr,

and A0i = B0

σ0(i), Arij = Br

σr(i)σir(j).

Given two points P and Q, we denote by d(P ,Q) the Euclidean distance betweenthem (in fact, we could use any other distance if necessary).

Definition 2.4. We define a distance on ΩL by

d(A,B) = infσ0∈ΣN0

N0∑i=1

d(A0i ,B

+s∑r=1

infσr∈ΣNr

Nr∑i=1

infσir∈Σdir

dir∑j=1

d(Arij ,B

Example. Let A0, A1, A2, A3 be four points such that three of them are not on thesame line in the Euclidean plane E2 associated to R2. Let (A,L) and (B,L′) be thetwo following similar interpolation schemes:

A =A1, . . . ,A1︸︷︷︸

6 times

;A2, . . . ,A2︸︷︷︸6 times

;A3, . . . ,A3︸︷︷︸6 times

;A0,A0,A0︸︷︷︸3 times

L =∂α(A1); ∂α(A2); ∂α(A3)︸︷︷︸

|α|62

; ∂α(A0)︸︷︷︸|α|61

B1, . . . ,B1︸︷︷︸

6 times

;B2, . . . ,B2︸︷︷︸6 times

;B3,B3,B3︸︷︷︸3 times

;B0, . . . ,B0︸︷︷︸6 times

L′ =∂α(B1); ∂α(B2)︸︷︷︸

|α|62

; ∂α(B3)︸︷︷︸|α|61

; ∂α(B0)︸︷︷︸|α|62

with Bi = Ai, 0 6 i 6 3. Obviously, the schemes are similar and d(A,B) =3d(A0,A3) 6= 0.

Remark. ΩL is an open subset in the Euclidean space of dimension N (the comple-mentary set is closed as the set of all the solutions of an equation). Thus, given anyA ∈ ΩL we are sure that there exists B ∈ ΩL close enough to A.

Now we are in good position to summarize the situation by the following chart:

(ΩL, Cs(Ω))L P∗N

H PNV R

(A, f ) LA[f ] HA[f ] HA[f ](M )

nodes andfunction

datafrom f

Hermitepolynomial

valueof the polynomial

where we can consider different components, for instance:

L[f ] : A ∈ ΩL −→ LA[f ] ∈ P∗N ,

LA : f ∈ Cs(Ω) −→ LA[f ] ∈ P∗N ,

HA = H LA : f ∈ Cs(Ω) −→ HA[f ] ∈ PN ,

H[f ] = H L[f ] : A ∈ ΩL −→ HA[f ] ∈ PN .

For example, the mapping LA is such that

f 7→L0[A][f ], 1 6 i 6 N0

∪Lrij[A][f ], 1 6 r 6 s; 1 6 i 6 Nr; 1 6 j 6 dir

Now given f ∈ Cs, we consider

V H[f ] : (A,M ) ∈(ΩL,Rd

)7→ HA[f ](M ) ∈ R

and its different partial derivatives, that we denote respectively by

• D – the partial derivative relatively to M ,

• DA – the partial derivative relatively to A,

• ∇A – the partial derivative relatively to A,

• ∇j,A – the partial derivative relatively to the jth component of A

(depending upon j, we will have ∇j,A = ∇0i,A or ∇j,A = ∇rij,A).

We have

∇j,AH[f ] =∇j,A

N0∑i=1

ϕ0i [f ]p0

+∇j,A

s∑r=1

Nr∑j=1

dir∑j=1

ϕrij[f ]prij

N0∑i=1

∇j,A(ϕ0i [f ]

N0∑i=1

ϕ0i [f ]∇j,A

s∑r=1

Nr∑j=1

dir∑j=1

∇j,A(ϕrij[f ]

s∑r=1

Nr∑j=1

dir∑j=1

ϕrij[f ]∇j,A(prij).

Thus, for ∇j,A = ∇0i,A, since the data are relatively linearly independent, we

obtain that ∇0i,A(ϕ0

i ) = 0 if i 6= j, ∇0i,A(ϕrij ) = 0 and

∇0i,A(ϕ0j

df (A0i )

= DA0if = DA0

(ϕ0i [f ]

)and for the same reason, for ∇j,A = ∇rij,A, we have ∇rij,A(ϕ0

` ) = 0, ∇rij,A(ϕr′i′j′) = 0

whenever r 6= r′ or i 6= i′ or j 6= j′ and ∇rij,A(ϕrij) = DArij(ϕrij [f ]).

We still have to evaluate the derivative of each of the basis functions.Going back to the definition of a partial derivative [7], we fix all the components

of A = A1,A2, . . . ,AN but one, say Aj , in order to evaluate ∇j,A.Let B = B1,B2, . . . ,BN with Bi = Ai for i 6= j. As ΩL is an open set,

there exists Bj 6= Aj such that B ∈ ΩL, Bj close enough to Aj . Let p0i [A], prij[A]

(respectively p0i [B], prij[B]) be the basis functions associated to A (respectively B).

(1) Let us first consider the case where Aj = A0j , 1 6 j 6 N0, thus Bj = B0

j .For Bj close enough to Aj , we have p0

j[A](Bj) 6= 0 (because p0j[A](Aj) = 1

and the polynomial basis functions are continuous). This implies that

p0j[B] =

p0j[A]

p0j[A](Bj)

because the right-hand side is a polynomial of degree N , which vanishes at all thepoints of B but Bj , where it takes the value 1 by construction.

Differentiating with respect to Bj we obtain

∇j,B(p0j[B]

)= −p0

j[A]×DB0

j(p0j[A])

(p0j [A](Bj))2

and evaluating this derivative at A,

∇0j,A(p0j[A]

)= −p0

j[A]×DA0j

(p0j [A]

Now, let i 6= j, 1 6 i 6 N0 and Ai = A0i . In this case we have

pi[B] = p0j[B] = p0

i [A]− p0j[A]× p0

i [A](Bj)

p0j[A](Bj)

Differentiating with respect to Bj = B0j , since p0

i [A] and p0j[A] are independent of

B0j , we get

∇j,B(p0i [B]

)= −p0

j[A]×∇j,B

(p0i [A](Bj)

p0j[A](Bj)

∇j,B

(p0i [A](Bj)

p0j[A](Bj)

(DB0j(p0i [A]))p0

j [A](Bj)− p0i [A](Bj)(DB0

j(p0j[A]))

(p0j[A](Bj))2

Evaluating at Aj and using the fact that we have a Lagrange basis, we obtain that thevalue at Aj of

∇j,A

(p0i [A](Bj)

p0j[A](Bj)

)is DA0

j(p0i [A]) and, therefore,

∇0j,A(p0i [A]

)= −p0

j[A]DA0j

(p0i [A]

Consider, finally, Ai = Ark`. In this case we have

pi[B] = prk`[B] = prk`[A]− p0j[A]×

prk`[A](B0j )

p0j[A](B0

∇j,B(pi[B]

)= −p0

j[A]×∇j,B

(prk`[A](B0

p0j[A](B0

In the same manner as above we obtain that

∇0j,A(prk`[A]

)= −p0

j[A]DA0j

(prk`[A]

(2) Let us now consider the second case, where Aj = Ark`, Bj = Brk` 6= Aj .

With similar but longer computations we can get the three following analogousresults:

∇rk`,A(prk`[A]

)=−prk`[A]×DAr

(ϕrk`[A]

(prk`[A]

∇rk`,A(p0i [A]

)=−prk`[A]DAr

(ϕrk`[A]

(p0i [A]

)), i 6= j, 1 6 i 6 N0,

∇rk`,A(ptij[A]

)=−prk`[A]×DAr

(ϕrk`[A]

(ptij[A]

)), (t, i, j) 6= (r, k, `).

We state all these results in the following lemma:

Lemma 2.1. Let i, j be two integers such that 1 6 i, j 6 N . Then for any polynomialpi[A] in the set of basis functions associated to A, we have

∇j,A(pi[A]

)= −pj[A] DAj

(ϕj[A]

(pi[A]

We can now proceed to evaluate

∇j,A(HAf ) = ∇j,A(Hf ).

We have

∇j,A(Hf ) =

N0∑i=1

−pj[A]DAj(ϕj[A]

(p0i [A]

))ϕ0i [A](f )

+s∑r=1

Nr∑j=1

dir∑j=1

ϕrk`[A](f )(−pj[ADAj

(ϕj[A]

(prk`[A]

)))+ pj[A] DAj

(ϕj[A](f )

that is,

∇j,A(Hf ) = pj[A]DAj(Lj[A](f )

)− pj[A]

N0∑i=1

ϕ0i [A](f )DAj

(ϕj[A]

(p0i [A]

s∑r=1

Nr∑j=1

dir∑j=1

ϕrk`[A](f )DAj(ϕj[A]

(prk`[A]

which can be written more simply as

∇j,A(Hf ) = pj[A] DAj(ϕj[A](f −HAf )

This can provide the value of all the partial derivatives with respect to A.For j such that Aj = A0

j , we have directly

DAjϕj[A](f −HAf ) = DA0j(f −HAf )

)= ϕA0

jj(D(f −HAf ) · v

More generally, let V and V ′ be two vector fields. From [7], we know that thederivative of the mapping M 7→ Drf (M ) · V r

M is the mapping V ′ 7→ Dr+1f (M ) ·(V r,V ′)M and from the Schwarz theorem for cross derivatives that

Dr+1f (M0) ·(V r,V ′

= D(Drf (M0) · V ′M0

)· V r

Thus we obtain

DAjϕj[A](f −HAf ) =DAj(DrAj (f −HAf ) · V r

)· V ′Aj

=DrAj(DAj (f −HAf ) · V ′Aj

)· V r

=ϕj[A](D(f −HAf ) · V ′

)and, taking V ′ = (V ′A1

,V ′A2, . . . ,V ′AN ), we get

∇A(Hf ) · V ′=N0∑i=1

p0i [A] ϕ0

i [A](D(f −HAf ) · V ′

s∑r=1

Nr∑j=1

dir∑j=1

prij[A]ϕrij[A](D(f −HAf ) · V ′

which is stated more simply as:

Proposition 2.2. The mapping

A ∈ ΩL 7→ HA[f ] ∈ PN

is differentiable and its derivative is defined by

∇A(H[f ]

)· V ′ = HA

[D(f −HA[f ]

)· V ′].

Let A = (A1,A2, . . . ,AN ) and B = (B1,B2, . . . ,BN ) be two interpolation setsin ΩL and denote by [A,B] ⊂ ΩL the segment line from A to B in ΩL, i.e., the setof the unisolvent sets C = (C1,C2, . . . ,CN ) in ΩL such that

[A,B] =C | ∃t ∈ [0, 1]: Ci = (1− t)Ai + tBi, 1 6 i 6 N

According to [7], the set ΩL is an open set, thus whenever [A,B] ⊂ ΩL we have∥∥HA[f ]−HB[f ]∥∥N ,∞,Ω 6 d(A,B) sup

C∈[A,B]

∥∥∇C(H[f ])∥∥N ,∞,Ω.

As Ω is bounded and non-empty, there exists a constant C such that for anyf ∈ Cs

supC∈[A,B]

∥∥HA[f ]∥∥N ,∞,Ω 6 C‖f‖s,∞,Ω.

Therefore, whenever s 6 N − 1 (which is always the case when dealing with anyinterpolation scheme which is not a multivariate Taylor interpolation of order N in asingle point)

supC∈[A,B]

∥∥∇C(H[f ])∥∥k,∞,Ω 6 C

∥∥(f −HC[f ])∥∥s,∞,Ω 6 C(1 + C)‖f‖k,∞,Ω.

Theorem 2.4. There exists a constant C such that, given two interpolation sets Aand B of points belonging to ΩL and such that they can be joined by a segment line[A,B] ⊂ ΩL, we have∥∥HA[f ]−HB[f ]

∥∥N ,∞,Ω 6 C(1 +C) d(A,B) ‖f‖s,∞,Ω.

Remark. In the particular case of multivariate Taylor interpolation sets of order Ndefined in one point, respectively A and B, we can use the following result fromWhitney [4]: Let E be a compact domain in Rd and T :A ∈ E 7→ TA a set ofpolynomials of degree 6 m and let E1 be an open set such that E ⊂ E1.

By its Taylor expansion, a function f ∈ Cm(E1) induces a set of polynomialsA ∈ E 7→ T kA[f ], ∀k 6 m. The following theorem studies the reverse situation.

Whitney’s theorem. A set of polynomials T :A ∈ E 7→ TA of degree 6 m is inducedby a function f ∈ Cm(E1) if and only if there exists a (concave) modulus of continuityω such that for all A and B in E

sup|α|6m

∣∣∂αTB(A)− ∂αTA(A)∣∣

ABm−|α| 6 ω(AB).

An equivalent form of this inequality is derived in [4]. Let A and B be twopoints such that [A,B] ⊂ E and let S(A,B) be the sphere of diameter AB, thenWhitney’s inequality can be replaced by∣∣TA(M )− TB(M )

∣∣ 6 1m!

(AMmω(AM ) +BMmω(BM )

Then, using Markov’s inequality, it easily follows that for the Taylor interpolation ofa function f in two points A and B of Ω one has∣∣∂αTNB [f ](M )− ∂αTNA [f ](M )

∣∣6 2|α|+1N2(N2 − 1) · · · (N2 − (|α| − 1)2)

[1 · 3 · · · (2α1 − 1)] · · · [1 · 3 · · · (2αd − 1)]1m!

ABN−|α| ω(AB)

for any M ∈ S(A,B).

3. On multivariate interpolation formulas

In this section we present some cases where the solution of an interpolationproblem can be easily obtained either explicitly or recursively. See also [10].

3.1. Multivariate Lagrange basis

Let N be an integer and

N = U(N , d) = dim(PN[Rd]).

Given a set of simple nodes A = (A1,A2, . . . ,AN ) in Ed, we are looking for an inter-polation formula similar to the one used for one-dimensional Lagrange interpolation,i.e.,

L[f ](M ) =N∑i=1

`i(M )f (Ai),

where the `i’s are polynomials of degree N such that `i(Aj) = δij , that is, 0 ifi 6= j and 1 if i = j. The existence of such a formula is obviously equivalent tothe unisolvency of the problem. Unfortunately, only in some special cases is it easilyavailable.

3.1.1. Chung–Yao interpolationLet us consider the following:

Geometric condition [3]. The set A = (A1,A2, . . . ,AN ) satisfies a geometric condi-tion if for any i = 1, . . . ,N there exist N hyperplanes Hi1,Hi2, . . . ,HiN such that:

(i) Ai /∈ Hik, for any k = 1, . . . ,N ,

(ii) any other Aj , j 6= i, belongs to at least one of the Hik, i.e., Aj ∈⋃Nk=1 Hik.

For simplicity, we also denote by Hik the polynomial of degree 1 that, up to aconstant factor, determines the hyperplane of the same notation.

For i = 1, 2, . . . ,N , let

`i(M ) =N∏k=1

Hik(M )Hik(Ai)

It is straightforward to see that `i: i = 1, 2, . . . ,N is the Lagrange basis for theinterpolation set A and the interpolating polynomial can be explicitly obtained as

PiNA [f ](M ) =N∑i=1

`i(M )f (Ai).

This formula implies the unisolvency of the problem.

3.1.2. Busch interpolationThis is an extension to Hermite interpolation of the Chung–Yao interpolation [2].Let n > 3 be an integer, N = U(N , d) the dimension of PN . Let A = (A1,

A2, . . . ,An) be the set of n distinct data points, each Ai with multiplicity mi (and, ofcourse,

∑ni=1mi = N ).

Let δβj [f ] = ∂βf (xj), for j = 1, . . . ,n and |β| 6 mi − 1.As usual, the interpolation scheme is a triplet (A,PN ,L), where L is the set

δβj [f ], j = 1, . . . ,n; |β| 6 mi − 1.

Hermite geometric condition. The set (A,PN ,L) satisfies a Hermite geometric con-dition if for any Ai ∈ A there exist ni = n−mi + 1 hyperplanes Hi1,Hi2, . . . ,Hini

such that:

(i) Ai /∈ Hik, for any k = 1, . . . ,ni,

(ii) any other Aj , j 6= i, belongs to at least one of the Hik, i.e., Aj ∈⋃nik=1 Hik.

As seen in [2], the problem is unisolvent and the Lagrange formula

ΠN [f ](M ) =n∑i=1

∑|α|6mi−1

`αi (M )∂αf (Ai)

can be constructed in a much more complicated form than in the simple case, exceptfor very simple particular cases.

3.1.3. Lagrange interpolation in the planeIn some particular examples, some simple geometric ideas can be useful in order

to construct the Lagrange basis.Let us consider first as an example the case N = 2 and let us assume that the

set A = (A1,A2,A3,A4,A5,A6) is P2-unisolvent. Geometrically, this means that thesix points are not on a cone.

In order to define `6, i.e., a polynomial of degree 2 which takes the value 1 atA6 and the value 0 for the five other points, we do the following:

• take four of the five points A1,A2,A3,A4,A5, say, A1,A2,A3,A4,

• consider two different sets of straight lines, each of them passing through two ofthese points, say (dAiAj is the line through A1 and A2)

γ1 = dA1A2dA3A4 and γ2 = dA1A3dA2A4 .

Now, let us consider any cone Γ12345 which goes through the points A1, . . . ,A5.It is well known that this cone is an element of the linear pencil of conic sectionspassing through A1,A2,A3,A4 and this pencil can be generated by γ1 and γ2. So letus write

Γ = Γ12345 = aγ1 + bγ2.

Proposition 3.1. Whenever A1,A2,A3,A4,A5 and A6 are not on the same conic sec-tion, there exists a unique couple a, b such that

Γ(A5) = aγ1(A5) + bγ2(A5) = 0,Γ(A6) = aγ1(A6) + bγ2(A6) = 1.

Then it is obvious that `6 = Γ12345 is the polynomial we were looking for.Similarly, we can construct the other `i’s.

For N = 3, let us assume that the set A = A1,A2, . . . ,A10 is P3-unisolvent.In order to define `10, we are seeking for the equation of a cubic curve Γ which wewrite as `10 = Γ with

Γ = a1Γ1 + a2Γ2 + a3Γ3,

where a1, a2, a3 are real numbers subject to

Γ(A8) = 0, Γ(A9) = 0, Γ(A10) = 1,

and Γ1, Γ2, Γ3 are three different cubes passing through the points A1,A2,A3,A4,A5,A6 and A7.

We can define them in the following manner: A1,A2,A3,A4,A5 define uniquelya cone γ1, and A6,A7 a unique line δ1. Now, let us set Γ1 = γ1δ1. Then, another

choice of five points out of the seven will give γ2 and with the two remaining pointswe have δ2, providing Γ2 = γ2δ2. Finally, another different choice of five and twopoints leads to Γ3 = γ3δ3.

The same idea can be extended to any N , forming N curves Γi = γiδi ofdegree N , with γi of degree N − 1 and δi of degree 1.

3.2. Newton recursive formula

Let us reformulate the usual univariate Newton formula in the following way.We are given a set A = x0,x1, . . . ,xN of N+1 different points. The (unique)

polynomial PNA [f ] of degree N interpolating a function f on A is defined recursivelyfor k = 0 up to N by

For k = 0, P 0x0[f ] = f0 = f (x0).

For k = 1, . . . ,N ,P kx0,x1,...,xk[f ](x)

= P k−1x0,x1,...,xk−1[f ](x) +

k−1∏i=0

(x− xixk − xi

)fk − P k−1

x0,x1,...,xk−1[f ](xk).

This can be extended to the multivariate case as follows.Let A = A0,A1, . . . ,AN be a PN [Rd]-unisolvent set of N distinct points.For any k = 1, . . . ,N , any Pk-unisolvent subset Ak of A contains at least one

subset Ak−1 ⊂ Ak ⊂ A which is Pk−1-unisolvent.Thus it is possible to define a nested sequence of unisolvent subsets of A,

A0 ⊂ A1 ⊂ A2 ⊂ · · · ⊂ AN−1 ⊂ AN = A.

If for each ai ∈ Ak \ Ak−1 we are able to construct the polynomial µki of degree ksuch that

µki (aj) = 0 if aj ∈ Ak \ Ak−1, aj 6= ai,µki (aki ) = 1,

then the following algorithm provides the polynomial PNA [f ] which interpolates f onA:

For k = 0 and A0 = a0, P 0A0

[f ] = f0 = f (a0).For k = 1, . . . ,N ,P kAk [f ](M ) = P k−1

Ak−1[f ](M ) +

∑ai∈Ak\Ak−1

µpi (M )f (ai)− P k−1

Ak−1[f ](ai)

In fact, P pAp[f ] is a polynomial of degree p which interpolates f on Ap, as iseasily seen when replacing M by any point of Ap.

Example. Let Σ be a triangle with vertices a1, a2, a3 and let a4, a5, a6 be three pointsdistinct from the vertices and respectively on the side opposite to a1, a2, a3.

Let A = a1, a2, a3, a4, a5, a6. From the Bezout decomposition, A is easily seento be P2-unisolvent.

Let us consider the nested decomposition

a1︸︷︷︸A1

⊂ a1, a2, a3︸︷︷︸A2

⊂ a1, a2, a3, a4, a5, a6︸︷︷︸A3

(1) Let P 0(M ) = f (a1).(2) We have

P 1(M ) = P 0(a1) +3∑i=2

µ1i (M )

f (ai)− f (a1)

µ1i (M ) = λi(M ), i = 1, 2, 3,

is the barycentric coordinate of M relative to ai in Σ. So we get

P 1(M ) = f (a1) +3∑i=2

λi(M )f (ai)− f (a1)

which is the usual formula for Lagrange interpolation at the vertices of a triangle.(3) At the next step, for N = 2, we obtain

P 2(M ) = P 1(a1) +6∑i=4

µ2i (M )

f (ai)− P 1(a1)

µ24(M ) = λ2(M )λ3(M ), µ2

5(M ) = λ3(M )λ1(M ), µ25(M ) = λ1(M )λ2(M ).

Now P 2 is the interpolating polynomial which solves the problem.We could have made another choice for the decomposition, for instance,

a1︸︷︷︸A1

⊂ a1, a6, a5︸︷︷︸A2

⊂ a1, a2, a3, a4, a5, a6︸︷︷︸A3

In this case, the intermediate polynomial P 1 would be the one interpolating over thetriangle [a1, a6, a5], but of course we would have ended up with the same polynomialbecause of the unisolvency.

3.3. Aitken-type interpolation in the plane

Let A = A0N = A1,A2, . . . ,AN be a set of N = U(N , 2) distinct points in

the plane defined recursively in the following way (see [27] and also [10]):

• There exist three different lines D1,1N ,D1,2

N ,D1,3N such that there are exactly N + 1

points of A lying on each line D1,iN . Let D1,i

N , i = 1, 2, 3, be the sets of these points.The lines are supposed to be neither parallel nor concurrent at one point.

• For each set A1,iN−1 = A0

N \D1,iN , i = 1, 2, 3, there exist three lines D1,i,j

N−1, j = 1, 2, 3

(with the same conditions as above) such that D1,i,jN−1 contains exactly N points of

the A1,iN−1. Let D2,i,j

N−1 be the set of N data points of D1,i,jN−1.

• And so on until there are three remaining points not on the same line.

Such sets of points evidently arise when dealing with a triangular set of data pointsdefined on the Bezier net of order N of a triangle [a1, a2, a3], that is, the net called byother authors (see [3], for example) principal lattices. These points are defined fromtheir barycentric coordinates α1/N ,α2/N ,α3/N , where |α| = α1 + α2 + α3 = N .

Another example is the triangular set of data points in [20], which are obtainedby the intersection points of three different pencils of N lines emanating from threevertices V1,V2,V3 not on the same line, in such a way that each data point Ak is oforder 3, i.e., there are three different lines, one from each pencil, which intersect atAk. The previous example of the Bezier net is a particular case of the second one,where the three vertices V1,V2,V3 are points at infinity, providing linear pencils ofparallel and equidistant lines.

So let us assume that we have an Aitken-type set of points A = A0N =

A1,A2, . . . ,AN , with the decomposition D1,iN , A1,i

N−1 = A0 \ D1,iN , i = 1, 2, 3.

Let A(i) = D1,i−1N ∩D1,i+1

N . From the construction, the points A(i), i = 1, 2, 3, arenot on the same line, thus define a proper triangle. Let us denote by λi the barycentriccoordinate associated with A(i) in this triangle, for i = 1, 2, 3.

It is easy to check that for an Aitken-type set of points, we have

=3∑i=1

λiPN−1A1,iN−1

where PkB denotes the polynomial of degree k defined by the interpolation set B.This can be iterated until we get final sets of points consisting of three non-

colinear points, providing a polynomial of degree 1.This algorithm can be defined in the same way in higher dimensions, but the

notation becomes complicated. We have to replace lines by hyperplanes, and trianglesby simplices.

3.4. Multivariate Abel interpolation

This particular problem can be characterized by the fact that we use only onepoint for each order of derivation, i.e., the problem can be posed in the following form:

Abel interpolation. Given a set of points A = Aα, |α| 6 N, find a polynomialPN ∈ PN [Rd] such that ∂αPN (Aα) = ∂αf (Aα) ∀α 6 N.

Obviously, the most important case is Taylor interpolation, where the points Aαall coalesce in A.

In the case of Abel interpolation, the Bezout decomposition theorem does notapply, but using incidence matrices [26], one can prove that:

Theorem 3.1 [23]. An interpolation scheme in Rd is regular if and only if it is anAbel interpolation scheme.

For Taylor interpolation around A, we know, of course, an explicit formula, i.e.,

TNA [f ](M ) =∑|α|6N

1α!∂αf (A)AMα,

but what about Abel interpolation in general?

3.4.1. Univariate caseLet A = x0,x1, . . . ,xN ⊂ R be a set of N + 1 points. Without loss of

generality, we can assume that x0 = 0.Now we can consider the following algorithm:

p1(x) = x− x0 = x.For k = 2, . . . ,N ,p(x;x1,x2, . . . ,xk−1)

=[p1(x)

]k − k−1∑i=1

)[p1(x)

]k−ipi(xk−i;xk−i+1, . . . ,xk−1).

It is easy to see that the polynomial Pk(x) = p(x;x1,x2, . . . ,xk−1) verifiesP (i)k (xi) = 0, i = 0, 1, . . . , k − 1, and so, if we define P0(x) = 1 the Lagrange

formula for Abel interpolation is

ΠNx0,x1,...,xN[f ](M ) =

N∑i=1

1k!f (k)(xk)Pk(x),

which coincides with the Taylor polynomial in the particular case where all the pointsare the same x0 = 0.

For any x0 we can make a simple change of variable.

3.4.2. Multivariate caseNow we take A = A0,A1, . . . ,AN and the interpolating conditions

∂γ(ΠN (A|γ|) = ∂γf (A|γ|)) for each multi-index γ with |γ| 6 N . Without loss ofgenerality, we assume that A0 = 0.

We define a finite sequence of polynomials Pγ by the following algorithm:

For |γ| = 1, pγ(x) = [OX]γ .For k = 2, . . . ,N

For |γ| = kpγ(X;A1,A2, . . . ,Ak−1)

=k!γ!

[OX]γ −k−1∑i=1

) ∑|α|=i

[OX]αpγ−α(Ai;Ai+1, . . . ,Ak−1),

where γ−α is the usual componentwise difference between multi-indices. If we define

P0(X) = 1 and Pγ(X) = pγ(X;A1,A2, . . . ,Ak−1) for |γ| = k,

the Lagrange formula for Abel polynomial interpolation can be written as

ΠNA0,A1,...,AN[f ](X) =

∑|γ|6N

1γ!∂γf (A|γ|)Pγ(X),

or, equivalently,

ΠNA0,A1,...,AN[f ](X) =

N∑k=0

∑|γ|=k

1γ!∂γf (Ak)Pγ(X)

(with the same remark as above for the case of general A0), which also reduces to theTaylor polynomial expansion in the particular case. Let us now give some examples.

Table of Abel polynomials of order 6 4

|γ|= 1:

p(1,0)(M ) = x,

p(0,1)(M ) = y,

|γ|= 2:

p(2,0)(M ,A1) = x2 − 2xp(1,0)(A1) = x2 − 2xx1,

p(1,1)(M ,A1) = 2xy − 2xy1 − 2yx1,

p(0,2)(M ,A1) = y(y − 2x1),

|γ|= 3:

p(3,0)(M ,A1,A2) = x3 − 3x2p(1,0)(A2)− 3xp(2,0)(A1,A2),

p(2,1)(M ,A1,A2) = 3x2y − 3[x2p(0,1)(A2) + 2xyp(1,0)(A2)]

− 3[xp(1,1)(A1,A2) + yp(2,0)(A1,A2)],

p(1,2)(M ,A1,A2) = . . . ,

p(0,3)(M ,A1,A2) = . . . ,

|γ|= 4:

p(4,0)(M ,A1,A2,A3) = x4 − 4x3p(1,0)(A3)− 6x2p(2,0)(A2,A3)

− 4xp(3,0)(A1,A2,A3),

p(3,1)(M ,A1,A2,A3) = 4x3y − 4[x3p(0,1)(A3) + 3x2yp(1,0)(A3)]

− 6[x2p(1,1)(A1,A2) + 2xyp(2,0)(A2,A3)]

− 4[xp(2,1)(A1,A2,A3) + yp(3,0)(A1,A2,A3)],

p(2,2)(M ,A1,A2,A3) = 6x2y2 − 12[x2yp(0,1)(A3) + xy2p(1,0)(A3)]

− 6[x2p(0,2)(A2,A3) + 2xyp(1,1)(A1,A2)

+ y2p(2,0)(A2,A3)]

− 4[xp(1,2)(A1,A2,A3) + yp(2,1)(A1,A2,A3)],

p(1,3)(M ,A1,A2,A3) = . . . ,p(0,4)(M ,A1,A2,A3) = . . . .

Remark. Some other important cases for practical construction of the solution of mul-tivariate polynomial Hermite interpolation problems arise when dealing with finiteelements, where the data points are strongly related to the structure of triangles, tetra-hedra or eventually d-simplices in Rd. Due to the lack of space, we refer to [16,18]where very efficients algorithms for evaluation of interpolating polynomials in thisspecific setting can be found.

References

[1] T. Bloom, On the convergence of multivariate Lagrange interpolants, Constr. Approx. 5 (1989)415–435.

[2] J.R. Busch, Osculatory interpolation in Rn, SIAM J. Num. Anal. 22 (1985) 107–113.[3] C.K. Chung and T.H. Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer.

Anal. 14 (1977) 735–741.[4] C. Coatmelec, Approximation et Interpolation des fonctions differentiables de plusieurs variables,

Ann. Sci. Ecole Norm. Sup. 83 (1966) 271–341.[5] P.J. Davis, Interpolation and Approximation (Blaisdell, New York, 1963).[6] C. de Boor and A. Ron, On multivariate polynomial Interpolation, Constr. Approx. 6 (1990).[7] J. Dieudonne, Fondements de l’Analyse Moderne (Gauthiers-Villars, Paris, 1967).[8] M. Gasca and A. Lopez-Carmona, A general recurrence formula and its applications to multivariate

interpolation, J. Approx. Theory 34 (1982) 361–374.[9] M. Gasca and J.I. Maetzu, On Lagrange and Hermite interpolation in Rn, Numer. Math. 39 (1982)

1–14.[10] M. Gasca and T. Sauer, Polynomial interpolation in several variables, this issue.[11] H.V. Gevorgian, H.A. Hakopian and A.A. Sahakian, On the bivariate Hermite interpolation problem,

Preprint, Arbeitspapiere der GMD (1992).[12] G. Glaeser, L’interpolation des fonctions differentiables, in: Proc. Liverpool Singularities Sympo-

sium II, Lecture Notes in Mathematics, Vol. 209 (Springer, Berlin, 1971) pp. 1–33.[13] R.B. Guenther and E.L. Roetman, Some observations on interpolation in higher dimensions, Math.

Comp. 24 (1970).[14] A. Le Mehaute, Taylor interpolation of order n at the vertices of a triangle. Application for Her-

mite interpolation and finite elements, in: Approximation Theory and Applications, ed. S. Ziegler

(Academic Press, New York, 1981).[15] A. Le Mehaute, On Hermite elements of class Cq in R3, Approximation Theory IV, eds. C.K. Chui,

L.L. Schumaker and J. Ward (Academic Press, New York, 1983).[16] A. Le Mehaute, Interpolation et approximation par des fonctions polynomiales par morceaux dans

Rn, These de Doctorat es-Sciences, Universite de Rennes (1984).[17] A. Le Mehaute, Unisolvent interpolation in Rn and the simplicial polynomial finite element method,

in: Topics in Multivariate Approximation, eds. C.K. Chui, L.L. Schumaker and F.I. Utreras (Acad-emic Press, 1987).

[18] A. Le Mehaute, An efficient algorithm for Ck simplicial finite element interpolation, in: MultivariateApproximation and Interpolation, eds. W. Haussman and K. Jetter (Birkhauser, Basel, 1990) pp.179–192.

[19] A. Le Mehaute, On multivariate Hermite polynomial interpolation, Multivariate Approximations,from CAGD to Wavelets, eds. K. Jetter and F. Utreras (1993) pp. 179–192.

[20] S.L. Lee and G.M. Phillips, Interpolation on the triangle and simplex, in: Approximation Theory,Wavelets and Applications, ed. S.P. Singh (Kluwer Academic, 1995) pp. 177–196.

[21] G.G. Lorentz and R.A. Lorentz, Solvability problems of bivariate interpolation, Constr. Approx. 2(1986).

[22] R.A. Lorentz, Uniform bivariate Hermite interpolation, in: Mathematical Methods in CAGD, eds.T. Lyche and L.L. Schumaker (Academic Press, Boston, 1989).

[23] R.A. Lorentz, Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, Vol. 1516(Springer, Berlin, 1992).

[24] S.H. Paskov, Singularity of bivariate interpolation, J. Approx. Theory 70 (1992).[25] Th. Sauer and Y. Su, On multivariate Lagrange interpolation, Math. Comp. 64 (1995) 1147–1170.[26] I.J. Schoenberg, On Hermite–Birkhoff interpolation, J. Math. Anal. Appl. 16 (1966).[27] H.C. Thacher, Jr. and W.E. Milne, Interpolation in several variables, J. SIAM 8 (1960) 33–42.[28] R.S. Walker, Algebraic Curves (Springer, New York, 1978).

on some aspects of multivariate polynomial interpolation

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