UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES On the error in multivariate polynomial interpolation

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. AMS (MOS) Subject Classifications: 41A63, 41A05; 41A10 Author’s affiliation and address: Center for the Mathematical Sciences University of Wisconsin-Madison 610 Walnut St. Madison WI 53705 1 supported by the National Science Foundation under Grant No. DMS-9000053 and by the United States Army under Contract No. DAAL03-90-G-0090 On the error in multivariate polynomial interpolation C. de Boor 1 Dedicated to Garrett Birkhoff on the occasion of his 80th birthday In interpolation, one hopes to determine, for g defined (at least) on a given pointset Θ, a function f from a given collection F which agrees with g on Θ. If, for arbitrary g, there is exactly one f ∈ F with f = g on Θ, then one calls the pair 〈F,Θ〉 correct. (Birkhoff [Bi79] and others would say that, in this case, the problem of interpolating from F to data on Θ is well set.) Assuming that F is a finite-dimensional linear space, correctness of 〈F,Θ〉 is equivalent to having (0.1) dimF = #Θ = dimF|Θ (with F|Θ := {f|Θ : f ∈ F} the set of restrictions of f ∈ F to Θ). 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 IR with d > 1 and 1 < dimF < ∞, there exist pointsets Θ ⊂ IR with dimF = #Θ > dimF|Θ. This observation rests on the following argument (see, e.g., the cover of [L66] or p.25 therein): For any basis Φ = (φ1, . . . , φn) for F , and any continuous curve γ : [0 d1] → (IR )n : t 7→ (γ1(t), . . . , γn(t)), the function g : t 7→ det(φj(γi(t))) is continuous. Since n > 1 and d > 1, we can so choose the curve γ that, e.g., γ(1) = (γ2(0), γ1(0), γ3(0), . . . , γn(0)), while, for any t, the n entries of γ(t) are pairwise distinct. Since then g(1) = −g(0), we must have g(t) = 0 for some t ∈ [0 d1], hence F is of dimension < n when restricted to the corresponding pointset Θ := {γ1(t), . . . , γn(t)}. As a consequence, it is not possible for n, d > 1 (as it is for n = 1 or d = 1) to find an n-dimensional space of continuous functions which is correct for every n-point set Θ ∈ IR. Rather, one has to choose such a correct interpolating space in dependence on the pointset. A particular choice of such a polynomial space ΠΘ for given Θ has recently been proposed in [BR90], a list of its many properties has been offered and proved in [BR9092], its computational aspects have been detailed in [BR91], and its generalization, from interpolation at a set of n points in IR to interpolation at n arbitrary linearly independent linear functionals on the space Π = Π(IR) of all polynomials on IR, has been treated in much detail in [BR92]. 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 [BR91-92], as part of more general results.