On the Basis Property of the System of Exponentials and Trigonometric Systems of Sine and Cosine Functions in Weighted Grand Lebesgue Spaces

IF 0.2 Q4 MATHEMATICS

Moscow University Mathematics Bulletin Pub Date : 2024-07-29 DOI:10.3103/s0027132224700128

M. I. Ismailov, I. F. Aliyarova

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引用次数: 0

Abstract

The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace \(G_{p),\rho}(a,b)\) of the weighted space of the grand Lebesgue space \(L_{p),\rho}(a,b)\) is defined. The density in \(G_{p),\rho}(a,b)\) of the set \(G_{0}^{\infty}([a,b])\) of infinitely differentiable functions that are finite on \([a,b]\) is studied. It is proved that if the weight function \(\rho\) satisfies the Mackenhoupt condition, then the system of exponentials \(\left\{e^{int}\right\}_{n\in Z}\) forms a basis in \(G_{p),\rho}(-\pi,\pi)\), and trigonometric systems of sine \(\left\{\sin nt\right\}_{n\geqslant 1}\) and cosine \(\left\{\cos nt\right\}_{n\geqslant 0}\) functions form bases in \(G_{p),\rho}(0,\pi)\).

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论加权大勒贝格空间中正弦和余弦函数的指数和三角函数系统的基础性质

摘要本文主要研究在移位算子产生的加权大勒贝斯格空间的可分离子空间中正弦和余弦函数的指数和三角函数系统的基础性质。本文借助移位算子，定义了大勒贝格空间的加权空间 \(L_{p),\rho}(a,b)\)的可分离子空间 \(G_{p),\rho}(a,b)\)。研究了在\([a,b]\)上有限的无限微分函数集合\(G_{0}^{\infty}([a,b])\)的密度。研究证明，如果权重函数 \(\rho\) 满足 Mackenhoupt 条件，那么指数系统 \(\left\{e^{int}\right\}_{n\in Z}\) 在 \(G_{p),\rho}(-\pi、\和余弦函数的三角函数系形成了 \(G_{p),\rho}(0,\pi)\) 中的基。

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来源期刊

Moscow University Mathematics Bulletin MATHEMATICS-

CiteScore

0.60

自引率

25.00%

发文量

期刊介绍： Moscow University Mathematics Bulletin is the journal of scientific publications reflecting the most important areas of mathematical studies at Lomonosov Moscow State University. The journal covers research in theory of functions, functional analysis, algebra, geometry, topology, ordinary and partial differential equations, probability theory, stochastic processes, mathematical statistics, optimal control, number theory, mathematical logic, theory of algorithms, discrete mathematics and computational mathematics.