关于多项式实零点个数的布洛赫-波利亚定理的扩展 (II)

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2024-11-15 DOI:10.1016/j.jat.2024.106122

Tamás Erdélyi

引用次数: 0

摘要

我们证明存在一个绝对常数 c>0，使得对于每一个 a0,a1,...,an∈[1,M],1≤M≤14expn9,有 b0,b1,...,bn∈{-1,0,1}，使得形式为 P(z)=∑j=0nbjajzj 的多项式 P 在 Ia 中至少有 cnlog(4M)1/2-1 个不同的符号变化：=(1-2a,1-a)，其中 a:=log(4M)n1/2≤1/3.这改进并扩展了布洛赫、波利亚和埃尔德利的早期结果，并作为一个特例，重现了雅各布和纳扎罗夫的一个更普遍的最新结果的特例。

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Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II)

We prove that there is an absolute constant

c > 0

such that for every

a_{0}, a_{1}, \dots, a_{n} \in [1, M], 1 \leq M \leq \frac{1}{4} exp (\frac{n}{9}),

there are

b_{0}, b_{1}, \dots, b_{n} \in {- 1, 0, 1}

such that the polynomial

P

of the form

P (z) = \sum_{j = 0}^{n} b_{j} a_{j} z^{j}

has at least

c {(\frac{n}{log (4 M)})}^{1 / 2} - 1

distinct sign changes in

I_{a} : = (1 - 2 a, 1 - a)

, where

a : = {(\frac{log (4 M)}{n})}^{1 / 2} \leq 1 / 3

. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.

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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.

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