Polynomial Interpolation in R3

K e y w o r d s L a g r a n g e interpolation, Simultaneous approximation, Freud weights 1. I N T R O D U C T I O N In this paper , we are concerned with construct ing in terpola t ing subspaces of polynomials of several variables of relat ively small dimension as well as the corresponding in terpola t ion formulae a l a Lagrange. DEFINITION 1. A (linear) subspace G C C(Rn) is called k-interpolating i f for every choice of distinct points ~l~U2~. . , u k E Rn, and for any choice of scalars el,a2, . , a k C R, there exists g 6 G such tha t g(ui) = ai, for all j = 1, 2 , . . . , k. We are interested in identifying kinterpola t ing subspaces G c C(Rn) with small dimensions. DEFINITION 2. Define I (k , n) = inf{dim G : G C C(R~) is k-interpolating}. A subspace G C C(P~) minimal k-interpolating f f it is k-interpolating and dim G = I(k, n). Obviously, for n = 1, we have I(k , 1) = k and the space Pk of polynomials of degree k I is a minimal in terpola t ing space. Very l i t t le is known in general abou t minimal in terpolat ing subspaces in C(Rn) for n > 1. Here, are a few known facts (1) I (2 , n) = n + 1, I(3, n) = n + 2, for all n > 1, (cf. [1,2]). (2) 2k 0(k) <_ I(k, 2) < 2k 1, where 0(k) is the number of l ' s in the b inary representat ion of the integer k. In par t icu lar I(4, 2) = 7. This remarkable inequal i ty was proved in [3] (cf. also [4]). (3) I(k , n) < (n + 1)k, for all k and n (cf. [5]). 0898-1221/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by ~4.MS-TEX doN: 10.1016/j.camwa.2004.10.022