The interpolation Problem for k-Sparse Sums of Eigenfunctions of Operators

In [DG 891, the authors show that many results concerning the problem of efficient interpolation of k-sparse multivariate polynomials can be formulated and proved in the general setting of k-sparse sums of characters of abelian monoids. In this note we describe another conceptual framework for the interpolation problem. In this framework, we consider R-algebras of functions &i, . . . , &” on an integral domain R, together with R-linear operators .LSi: 4 + 4. We then consider functions f from R” to R that can be expressed as the sum of k terms, each term being an R-multiple of an n-fold product f&xi) . * * . *f&J, where each fi is an eigenfunction for ~3~. We show how these functions can be thought of as k-sums of characters on an associated abelian monoid. This allows one to use the results of [DG 891 to solve interpolation problems for k-sparse sums of functions which, at first glance, do not seem to be characters. Let R, &‘i, . . . , JZ&, and LSi,. . . , g,, be as above. For each A E R and 1 I i I n, define the A-eigenspace g^ of Bi by