Chebyshev systems and Sturm oscillation theory for discrete polynomials

TL;DR

This paper extends classical approximation and oscillation theorems to discrete functions, establishing Chebyshev systems and Sturm oscillation results for discrete polynomials, with applications to orthogonal polynomials and spectral gap problems.

Contribution

It introduces discrete analogues of Chebyshev's alternation theorem and Sturm's oscillation theorem, and applies these to orthogonal polynomials and extremal problems.

Findings

Discrete Chebyshev systems characterized by alternance sets.

Number of zeros of discrete polynomials bounded by indices, confirming Sturm's theory.

Monotonicity of Fourier coefficients in orthogonal polynomials with removed zeros.

Abstract

We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $\Phi_n=\{\varphi_k\}_{k=1}^n$ on the interval $[0,q]_{\mathbb{Z}}=[0,q]\cap \mathbb{Z}$. In particular, we establish that the polynomial of best uniform approximation of a discrete function $f$ admits a Chebyshev alternance set of length $n+1$ if and only if $\Phi_n$ is a Chebyshev $T_{\mathbb{Z}}$-system. Also, we obtain a discrete version of Sturm's oscillation theorem, according to which the number of discrete zeros of the polynomial $\sum_{k=m}^{n}a_k\varphi_k$ is no less than $m-1$ and no more than $n-1$. This implies that $\Phi_n$ is a $T_{\mathbb{Z}}$-system and a discrete Sturm-Hurwitz spectral gap theorem is valid. As applications, we study the orthogonal polynomials with removed largest zeros. We establish the monotonicity property of coefficients in the Fourier expansions of such polynomials, thereby strengthening the results of H. Cohn and A. Kumar. We apply this to solve a Yudin-type extremal problem for polynomials with spectral gap.