Strictly positive definite functions of finite orders and multivariate polynomial interpolation

Abstract

We study strictly positive definite functions of finite orders on real inner product spaces. Via exploring new connections to theories of Chebyshev ranks and Schur polynomials, we obtain quantifiable conditions for such functions. To assist field scientists in selecting an ideal multivariate polynomial interpolant, we propose and study the notions of “interpolating rank” and “minimal seminorm interpolation.” We quantify both Chebyshev ranks and interpolating ranks in terms of the number of common zeros of a certain finite collection of Schur polynomials. As a byproduct, we develop a mechanism to construct positive-rank Chebyshev systems of which examples are scarce in the literature.

Department(s)

Mathematics

Document Type

Article

DOI

10.1016/j.jat.2025.106283

Keywords

Chebyshev ranks, Interpolating ranks, Multivariate polynomial interpolation, Positive definite functions, Schur polynomials

Publication Date

5-1-2026

Journal Title

Journal of Approximation Theory

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