关于有限制零点的多项式的反向马尔可夫-尼克尔斯基不等式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2024-08-30 DOI:10.1016/j.jat.2024.106097

Mikhail A. Komarov

引用次数: 0

摘要

设 Πn 是 n 阶代数多项式 P 的类，其所有零点都位于线段 [-1,1] 上。1995 年，S. P.周证明了下面的图兰型逆马尔科夫-尼克尔斯基不等式：P′‖Lp[-1,1]>c(n)1-1/p+1/q‖P‖Lq[-1,1]，P∈Πn，其中 0<p≤q≤∞，1-1/p+1/q≥0（c>0 是与 P 和 n 无关的常数）。我们还讨论了一些相关的图兰式不等式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros

Let $Π_{n}$ be the class of algebraic polynomials $P$ of degree $n$ , all of whose zeros lie on the segment $[- 1, 1]$ . In 1995, S. P. Zhou has proved the following Turán type reverse Markov–Nikol’skii inequality: ${‖ P^{'} ‖}_{L_{p} [- 1, 1]} > c {(\sqrt{n})}^{1 - 1 / p + 1 / q} {‖ P ‖}_{L_{q} [- 1, 1]}$ , $P \in Π_{n}$ , where $0 < p \leq q \leq \infty$ , $1 - 1 / p + 1 / q \geq 0$ ( $c > 0$ is a constant independent of $P$ and $n$ ). We show that Zhou’s estimate remains true in the case $p = \infty$ , $q > 1$ . Some of related Turán type inequalities are also discussed.

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.