Applied Numerical Mathematics 10 (1992) 297-305 North-Holland

297

APNUM346

C. de Boor Center for the Mathematical Sciences, University of Wisconsin-Madison, 610 Walnut Street, Madison, WI 53705, USA

Abstract

De Boor, C., On the error in multivariate polynomial interpolation, Applied Numerical Mathematics 10 (1992) 297-305.

Simple proofs are provided for two properties of a new multivariate polynomial interpolation scheme, due to Amos Ron and the author, and a formula for the interpolation error is derived and discussed.

Introduction

In interpolation, one hopes to determine, for g defined (at least) on a given point set 0, a function f from a given collection F which agrees with g on 0. If, for arbitrary g, there is exactly one f~ F with f = g on 0, then one calls the pair (F, 0) correct. (Birkhoff [l] and others would say that, in this case, the problem of interpolating from F to data on 0 is well set.) Assuming that F is a finite-dimensional linear space, correctness of (F, 0) is equivalent to having

(0 1) . dim F = #O = dim F[o

(with FIB := if lo: f E F) the set of restrictions of f E F to 0). Multivariate interpolation has to confront what one might call “loss of Haar”, i.e., the fact

that, for every linear space F of continuous functions on Rd with d > 1 and 1 c dim F < 00, there exist point sets 0 c IFid with dim F = #@ > dim Fle. This observation rests on the following argument (see, e.g., [S, the cover or p. 251): For any basis @ = Q,, . . . ,&) for F, and any continuous curve

y : [Q..l] + (UP)” : t I+ (r&)9 l l l 9 r,(%

Correspondence to: e. de Boor, Center for the Mathematical Sciences, University of Wisconsin-Madison, 610 Walnut Street, Madison, WI 53705, USA. * This work is supported by the National Science Foundation under Grant No. DMS-9000053 and by the United

States Army under Contract No. DAAL03-90-G-0090.

0168~9274/92/$05.00 0 I992 - Elsevier Science Publishers B.V. All rights reserved

298 C. de Boor / Multivaria te polynomial interpolation

the function g : t I+ det(&(yj( t))) is continuous. Since n > 1 and d > 1, we can SO choose the curve y that, e.g., ~(1) = (y,(O), r,(O), y&O), . . . , y,(O)), while, for any t, the n entries of y(t) are pairxjse distinct. Since then g(l) = -g(O), we must have g(t) = 0 for some t E [O..l], hence F is of dimension < n when restricted to the corres;ocding point set @ := ( yl(t), . . . , yn( t 1).

As a consequence, it is not possible for n,d > 1 (as it is for y1 = 1 or d = 1) to find an n-dimensional space of continuous functions which is correct for every n-point set 0 E Rd. Rather, one has to choose such a correct interpolating space in dependence on the point set.

A particular choice of such a polynomial space L+!@ for given 0 has recently been proposed in [2], a list of its many properties has been offered and proved in [2-41, its computational aspects have been detailed in [3], and its generalization, from interpolation at a set of yt points in Rd to interpolation at yt arbitrary linearly independent linear functionals on the space

n = n( IlP)

of ai1 polynomials on Iw d, has been treated in much detail in [4]. The present short note offers some discussion concerning the error in this new polynomial

interpolation scheme, and provides a short direct proof of two relevant properties of the interpolation scheme, whose proof was previously obtained, in [3,4], as part of more general results.

la. The interpolation scheme

To recall from [2,3], the interpolation scheme is stated in terms of a pairing. between LI and the space A, of all functions on Iwd with a ccnvergent power series (in fact, the larger space 67’ of all formal power series would work as well). Here is the pairing:

(1 1) . (p, s> :=p(D)g(O) = c Dap(0) D’%(O)/d. aEz.4

The sum is over all multi-indices ti, i.e., over all d-vectors with nonnegative integer entries, D* denotes the partial derivative Dye) - 0 l Ditd), and CU! :=r CW( l)! - l l a(d)!. Further, we will use the standard abbreviation

Id :=Q!(l)+ l ‘* ++), CYEZ”,,

and the nonstandard, but convenient, notation

();I: Rd + [w: x f+_p :=$) a(l) . . . +q”‘d’

ower function or cu-monomial. E.g., if p is the polynomial C,c(a)( )*, then p(D) is the constant coefficient differential operator C,c(a)D”.

The pairing is set up so that the linear functional

S(G): 17-3 &J:p hDp(6’)

of evaluation at 8 is represented with respect to this pairing by the exponential with frequency 8 E Rd, i.e., by the function

C. de Boor / Kultiuariate polynomial interpolation 299

(with 8x :== Cje(j),(j) tht usual scalar product). Indeed, since IYe, = 0”, one computes

(1 2) . (p, ee) = Cll”p(O)e*/~~! =p(e). a

Further, the pairing is graded, in the following sense. For g GA,-, 1 I7 and k E Z +, denote by

gik] := c D*g(O)( )“/a! ICrl=k

its kth-order term, i.e., the sum of all terms of exact order k in its power series expansion. In these terms, ptk] interacts in ( p, g) only with the corresponding gIk]. In particular, if we denote by pT the leading term of p E II, i.e., the nonzero term pIk] with maximal k, then

(PpP)=(Pt,PrDo,

except when p = 0, in which case, by convention, pT := 0. Correspondingly, we denote by g, the least or initial term of g Q,, i.e., the nonzero term gik] with minimal k, and conclude correspondingly that

(g,, 9) =h,9 g,)w

except when g = 0, in which case, by convention, g, :-- 0. Set now (as in [2])

Iif@ := sparrIg*: g E Exp,) with Exp := span(e,: 8 E 0).

Then II@ + Iwo : p -p 1 o is l-l: For, if ply = 0, then by (1.2), ( p, g) = 0 for all g E Exp,. If now p # 0, then necessarily pt = g i for some g E Exp, \ 0,

o=(p,g)=(P,,g,)=(P,,P,)>o,

a contradiction. It follows that dim & = dim(L!& 6 #@. On the other hand, it is possible to show, by a variant of

and then

the Gram-Schmidt orthogonaliza- tion process started from. the basis (eel0 E o for Exp,, the existence of a sequence (gr,. . . , g,) in Exp, (with y1 := #O), for which

(1 3) . (gil, gj) =O e i#j.

This shows, in particular, that (g, J,. . . , g,rl ) is independent (and in I&), hence that dim & 2 n=#@.

Consequently, (&, 0) is correct. More than that, for arbitrary f E 17,

is the unique element in ITo which 6 t @-cc&s tiit!z J a’c @. Indeed, it follows from (1.3) that, for i = l,..., n9 (I,f, gi> = (f, Si>* Since (&, $52,. . . 9 g,) is independent (by (1.3)), hence a basis for Exp, (as Exp, is spanned by n elements), we conclude with (1.2) that, for all 8 E 0,

r,f(e) = (I,f, e,> = (f, e,> =f(e).

300 C. de Boor / Multiuaria te yolynomial interpolation

mark. Since Exp,, as the span of n = #@ functions, is trivially of dimension < #@, we seem to have just proved that

(1% . dim Exp, = #O.

This is misleading, though, since the proof of the existence of that sequence (g,, g,, . y *, g,) in Exp, satisfying (1.3) uses (1.5).

Note that, with gj =: C, E o B( j, 8)e,,

(16) . (f, gj) = C WY @fW BE@

Thus, with (1.6) as a definition for ( f, gj) in case f E I7, (1.4) provides the polynomial interpolant from Z7, at Q to arbitrary f defined (at least) on 0.

2. Simple proofs of some properties of I,

As shown in [2-41, the interpolation scheme I, has many desirable properties. Some of these follow directly from the definition of TIT,: For example, II, c ZZel in case 0 c 0’ (leading to a Newton form for the interpolant). Also, for any Y > 0 and any c E Rd, ZZro+c = Z7@. The translation-invariance, ZZ@ + c = ZZ@, implies that II, is D-invariant, i.e.,

Further, for any invertible matrix C, ZZco = Z& 0 CT (with CT the transposed of C). Also, ZZ@ depends continuously on 0 (to the extent possible, limits on this being imposed by “loss of Haar”), and I, converges to appropriate Hermite interpolation if elements of @ are allowed to coalesce in a sufficiently nice manner.

Perhaps the two most striking properties are that

(i) Z@ is degree-reducing, and (ii) Z& = n pls=o ker p *CD).

These properties are proved in [2-4] as part of more general results. Because of the evident and expected importance of these results, it seems uLefu1 to provide direct proofs, which I now do.

The minimum-degree property:

(2 2) . V(P E n) deg I,,p G deg p,

follows immediately from (1.4) since ( p, gj > = 0 whenever deg p c deg gj, . (It is stressed in [l] that univariate Lagrange interpolation has this property.)

In fact, the inequality in (2.2) is strict if and only if pT _L &, as will be established during the proof of the second property. Here and below, I find it convenient to write p _L G (and say that “p is perpendicular to G”) in case ( p, g) = 0 for all g E 6, with p E I7 and G CA,.

C. de Boor / Multivaria te polynomial interpolation 301

(2.3) Proposition (de Boor and Ron [3,4]). I& = n D,_O ker p ,(D).

roof. I begin with a proof of the following string of equivalences and implications:

(2 4) . plo=Q * pi-Exp,

--7 p+n,

* v(s E II,) P f (D)4(0) = O

The first equivalence follows from (1.2), the second relies on the definition of orthogonality, and the third uses the fbcts that p(D)D” = D”p(D) (for any p E l7), and that the polynomial p T(D)q is the zero polynomial iff all its Taylor coefficients are zero.

The “ =+ ” follows from the observation that if (p, g) = 0, then ( p7, g,) = 0, either because deg pT # deg g, , or else because, in the contrary case, ( p, g) = ( pT , g, ). Finally, the “ = ” is trivial.

The “ = ” can actually be replaced by “ * ” since I7@ is D-invariant, by (2.1). Also, the “ * ” can be reversed in the following way:

For, if f is a polynomial perpendicular to &, of degree k say, then I,f is necessarily of degree < k, since, in the formula (1.4), the terms ( f, gj) for deg gj, > k are trivially zero while, for deg gj, = k, we have ( f, Sj) = ( f, gj, ) and this vanishes since f 1 He. Conse- quently p := f - I,f is a polynomial with the sam ‘s leading term as f and perpendicular to Expe-

In any case, the argument given so far shows that I& c n ple=o ker p ,(D). To show equality, note that dim L!@ = #O < 00, hence I7@ c hfk for some k. Thus, for any I a I = k + 1,

- deg I,( )U < deg( )a = k + 1,

hence

(( r - I@( )“), = ()“’ therefore n PIl_j=o ker pt(Dk n Ial=k+l ker Da = Ilk c II. Further, if q E II, then p := q - I@q is a polynomial of degree < deg q (by (2.2)) and p ,(D)(&) = 0 (since pie = O), therefore pT(D)p =p &D)q. Hence, if q E n Ple=o ker p ,(DJ, then p ,(D)p = 0, hence p = 0, i.e., q E n,. 0

3. Error

The standard error formula for univariate polynomial interpclation is based on the Newton form, i.e., on the “correction” term [or, 8,, . . . , IT?,, xl f l-If= ,I l - xi) which is added to the

302 C. de Boor / Multivariate polynomial interpolation

polynomial interpolating to f at 0,, 8,, . . . , 8, in order to obtain the polynomiai interpolating to fat e,, b.-, Ok, x. An analogous formula is available for the error f - I, f in our multivari- ate polynomial interpolant. For its description, it is convenient to use the respect to the pairing (l.l), i.e., the map

~~:A()-ul():g~ cg.

(gjJ9 g>

j '(giJ9 gj) l

(3.1) Proposition. For any x E Rd and any f E n,

(3 2) . f(x) - (I,f )(x) = (f, 4

with

(3 3) (R j.J 9 e,> . E@,~ := e, -Igex=e,- Lg.

j ‘(giJ9 gi) l

roof. Since e, represents the linear functional S(X) of evaluation at x with respect to (Ll), Z$e, is the exponential which represents the linear functional S( x)1@ with respect to (1.1). q

(3.4) Corollary. The exponential Q, x , represents S(x) on the ideal

ideal(O) := ker I, = (f E II: f 10 = 0),

and is orthogonal to I&, hence so are alk its homogeneous compcrrents E$,?.

roof. If f 10 = 0, then f(x) = f(x) - I@f(x> = ( f9 E@,~). Since Zg is the dual to the linear projector of interpolation from L!@, its interpolation

conditions are of the foim ( p9 l ) with p E I&. Hence E@,~, as the error e, - 1$e,, must be perpendicular to ?7@, ant; this, incidentally, can also be written as

p(D)q&) = 0 ‘dp E n, l

Finally, since He is spanned by homogeneous polynomials, f A_ IlO implies that f Ikl I l7,, for all i&Z+. 0

mma (de Boor and Ron [2]). For any x e 0, l7@ U x = l& + span{ p, J, with 3

(3 6) . 13o,x := ( e, -&X)l

the initial term of E@,~.

Proof. First, po,x E De” x since it is the initial term of some element of Exp, UX. Further, P@,~ Z 0 since P@,~ = 0 would imply that e, E Exp,, hence x E 0 by (i.5). Therefore

(3 7) . O<(Pg,*,Po,x)=(Po.x,Eo,x)=(Pe.-~~P~,,,e,)=(P,,,-I,P,.)(x), 9 9 showing that po,x - Z@P~,~ # 0, hence po,x E I&. q

C. de Boor / Multivaria te polynomial interpolation 303

C3.8) Corollary. For any ordering 0 = (0 1, O,, . . . , tI,,> and with @j := (0 ], 9,) . . . , e,>,

(3 9 . n (f9 go,>

j=l (PO,-I,,_l'@j) (p

03 p

0 i’

I I

Proof. The proof is by induction on #O, starting with the case y1= 0, i.e., 0 = ( ), for which the definition 4) := 0 is suitable. For any finite 0 and x e 8 and any f j we know from Lemma 3.5 that

(3.10) (f

P := I,f + (PO,, - I,P,,,) (p ’ &Bdx >

0,X) 0,x )

is in &,,, and from (3.7) and Propo sition 3.1, that p(x) = f( x), while evidently p = f on 0, hence p must be the polynomial I,, x f. ‘Thus if (3.9) holds for 0, it also holds for 0 U x. q

Such a Newton form for I,f was derived in a somewhat differtint manner in [a]. Note that

40,x := (P 0,x - I@P@,xV( P*,xs P&x)

is the unique element of He u x which vanishes at 0 and takes the value 1 at x. But there does not appear to be in general (as there is in the univariate case) a scaling sqe,x which makes its coefficient ( f, E@,,)/s in (3.10) independent of the way 0 ux has been split into 0 and X. The only obvious exception to this is the case when I7@ = & := the collection of polynomials of total degree < k. Thus, only for this case does one obtain from I* a ready multivariate divided difference.

Unless the ordering (8,, B,, . e . , 13~) is carefully chosen (e.g., as in the algorithm in [3]), there is no reason for the corresponding sequence (deg p8,, . . . , deg ps,) to be nondecreasing. En particular, deg pg,x may well be smaller than deg I,f. For example, if x is not in the affine hull of 0, then deg po,x = 1. This means that the order of the interpolation error, i.e., the largest integer k for which f(x) - I, f(x) = 0 for all f E l7, k, may well change with X, since it necessarily equals deg P@,~. The only exception to this occurs when I& = J.., for some k. More generally, deg P@,~ is a continuous function of X, hence constant, in some neighborhood of the point 5 if the point set 0 U 6 is regular in the sense of [2], i.e., if

for some k. (To be precise, de Boor and Ron [2] call Exp, u5 rather than 0 U 5 regular in this case.)

(3.11) Proposition. k := deg ps x , = min(deg p: p(x) + 0, p E ideal(@)).

Proof. Let

k’ := min(deg p: p(x) z 0, p E ideal(@)).

If p E ideal(@), then p(x) = ( p, E@,~) by Corollary 3.4, therefore p(x) f 0 implies k < dw P-

Thus k < k’. On the other hand, q := ps x - I,p,,, has degree \< deg P@,~ (by (2.2); in fact, we already know from the proof of Proposition 2.3 that deg q = deg P@,~, but we don’t need that

here) and does not vanish at X, by (3.7), hence also k’ < deg q G k. •I

304 C. de Boor / MultiEaria te polynomial interpolation

The derivation from (3.2) of useful error bounds requires suitable bounds for expressions like

in terms of norms like C, a, =k 11 D*f ]l(L,(B)), with k = deg p. x and B containing @ u X. Presumably, one would first shift the origin to lie in B, in order ‘to keep the constants small, and so as to benefit from the fact that E@,~ vanishes to order k at 0.

In view of Proposition 2.3, integral representations for the interpolation error f - 1J should be obtainable from the results of K. Smith, [6], using as differential operators the collection p ,(D), p E P, with P a minimal generating set for ideal( 0).

4. A generalization and Birkhoff% ideal interpolation schemes . .

In [I], Birkhoff gives the folloGng abstract description of interpolation schemes. With X some space of functions on some domain T into some field F and closed under pointwise multiplication, and Qi a collection of functionals (i.e., F-valued functions) on X, there is associated the data map

(for which Birkhoff uses the letter a). Birkhoff calls any right inverse I of S(G) an interpola- tion scheme on Qr. (To be precise, Birkhoff talks about maps I : Q, +X which are to be right inverses for S(a), and uses FY with Y c T as an example for @, but the intent Ss clear.) He observes that P := IS(@) is necessarily a projector, i.e., idempotent.

He calls the pair (S( @), I), or, better, the resulting projector P := IS( CD), an ideal interpola- tion scheme in case

(i) S(Q)1 = id; (ii) both S( @) and I are linear (hence P is linear);

(iii) ker P is an ideal, i.e., closed under pointwise multiplication by any element from X.

For linear S( @) and I: (is( @), I) is ideal if and only if ker S(a) is an ideal (since ker P = ker Ei(@) regardless o’ I). Thus any linear scheme for which the data map is a rezlriction map +f]e (such as the map I, discussed in the preceding sections) is trivially ideal,

In these terms, the generalization of I, treated in [4] deals with the situation when T = Rd and X=n=n<lR”>, and @:f++#$-)+-cp for some finite, linearly independent, collection of linear functionals cn n (with a further extension, to infinite @, also analyzed). The algebraic dual LV can be represented by the space of formal power series (in d indeterminates), and the pairing (1.1) has a natural extension to n x l7’.

In this setting,

ker S(@)=AL:=(pEn: p-LA),

with

A := span @.

, . 5

L. de Boor /’ Muitiuariate polynomial interpofaiion 305

(4.1) Proposition (de Boor and Ron [4]). ker S(Q) is an ideal if and only if A is D-invariant.

The proof uses nothing more than the observation that

<()“P, 4) = (P, DV>~

As an example, if @ is a linearly independent subset of U OE @e,l;T, then C$ E 43 is of the form

f c-)P(WfW

for some 8 E 0 and p E K Correspondingly, A = span Q, = C 8 E &$ for certain polynomial spaces PO. Hence, ker S(@) is an ideal iff each PO is D-invariant. In particular, Hermite interpolation at finitely many points is ideal, while G.D. Birkhoff interpolation is, in general, not.

References

[l] G. Birkhoff, Th e algebra of multivariate interpolation, in: C.V. Coffman and G.J. Fix, eds., Constructive Apprtiaches to Mathematical Models (Academic Press, New York, 1979) 345-363.

[2] C. de Boor and A. Ron, On multivariate polynomial interpolation, Constr. Approx. 6 (199C) 287-302. [3] C. de Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp. (to

appear). [4] C. de Boor and A. Ron, The least solution for the polynomial interpolation problem, Math. 2. (to appear). [5] G.G. Lorentz, Approximation of Functions (Holt, Rinehart and Winston, New York, 1966). [6] K.T. Smith, Formulas to represent functions by their derivatives, Math. Ann. 188 (1970) 53-77.