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Chebyshev systems and Sturm oscillation theory for discrete polynomials

D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov

TL;DR

The work develops a comprehensive discrete analogue of Chebyshev alternation and Sturm oscillation theory for polynomials on the integer grid $[0,q]_{\mathbb{Z}}$, introducing $T_{\mathbb{Z}}$-systems and linking them to best uniform approximation on discrete sets. It proves that a discrete best approximant has a Chebyshev alternance of length $n+1$ if and only if the generator set is a $T_{\mathbb{Z}}$-system, and establishes a discrete Sturm oscillation theorem bounding the zeros of discrete polynomial combinations. The authors then apply these results to orthogonal polynomials with removed largest zeros, proving monotonicity of Fourier coefficients in their expansions and solving a Yudin-type extremal problem for polynomials with spectral gap, with implications for spherical codes and designs. By connecting discrete Chebyshev systems to Sturm-Liouville theory and discrete spectral-gap phenomena, the paper provides new tools for discrete approximation and spectral analysis on finite integer grids.

Abstract

We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $Φ_n=\{\varphi_k\}_{k=1}^n$ 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 $Φ_n$ 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 $Φ_n$ 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.

Chebyshev systems and Sturm oscillation theory for discrete polynomials

TL;DR

The work develops a comprehensive discrete analogue of Chebyshev alternation and Sturm oscillation theory for polynomials on the integer grid , introducing -systems and linking them to best uniform approximation on discrete sets. It proves that a discrete best approximant has a Chebyshev alternance of length if and only if the generator set is a -system, and establishes a discrete Sturm oscillation theorem bounding the zeros of discrete polynomial combinations. The authors then apply these results to orthogonal polynomials with removed largest zeros, proving monotonicity of Fourier coefficients in their expansions and solving a Yudin-type extremal problem for polynomials with spectral gap, with implications for spherical codes and designs. By connecting discrete Chebyshev systems to Sturm-Liouville theory and discrete spectral-gap phenomena, the paper provides new tools for discrete approximation and spectral analysis on finite integer grids.

Abstract

We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions on the interval . In particular, we establish that the polynomial of best uniform approximation of a discrete function admits a Chebyshev alternance set of length if and only if is a Chebyshev -system. Also, we obtain a discrete version of Sturm's oscillation theorem, according to which the number of discrete zeros of the polynomial is no less than and no more than . This implies that is a -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.
Paper Structure (12 sections, 31 theorems, 177 equations)

This paper contains 12 sections, 31 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Discrete Chebyshev system
  3. Sturm oscillation theorem, $T$-systems, and orthogonal polynomials
  4. Monotonicity property of coefficients of polynomials with removed largest zeros
  5. An extremal problem for polynomials with spectral gap
  6. Structure of the paper
  7. Proofs of Theorem \ref{['theo3']} and Corollary \ref{['cor2']}
  8. Proofs of Theorems \ref{['theo4']}, \ref{['theo5']}, and \ref{['theo6']}
  9. Proof of Theorem \ref{['theo6']}
  10. Proof of Theorem \ref{['theo7']}
  11. Proof of Theorem \ref{['theo8']}
  12. Appendix

Key Result

Theorem 1.1

The best uniform approximant $p^*\in L_n$ of the function $f\in C[0,q]_{{\mathbb{Z}}}$ is unique if and only if the set $\{\varphi_k\}_{k=1}^{n}\subset C[0,q]_{{\mathbb{Z}}}$ is a $T_0$-system.

Theorems & Definitions (50)

  • Theorem 1.1: La72
  • Proposition 1.2
  • Theorem 1.3: La72
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9: Si05
  • Theorem 1.10
  • ...and 40 more