Convergence of General Fourier Series of Differentiable Functions

Pub Date : 2023-12-28 DOI:10.3103/s1068362323060067

V. Tsagareishvili

引用次数: 0

Abstract

Convergence of classical Fourier series (trigonometric, Haar, Walsh, \(\dots\) systems) of differentiable functions are trivial problems and they are well known. But general Fourier series, as it is known, even for the function \(f(x)=1\) does not converge. In such a case, if we want differentiable functions with respect to the general orthonormal system (ONS) \((\varphi_{n})\) to have convergent Fourier series, we must find the special conditions on the functions \(\varphi_{n}\) of system \((\varphi_{n})\). This problem is studied in the present paper. It is established that the resulting conditions are best possible. Subsystems of general orthonormal systems are considered.

查看原文

可微分函数的一般傅里叶级数的收敛性

摘要可微分函数的经典傅里叶级数（三角函数、哈氏函数、沃尔什函数、(\dots\)系统）的收敛是微不足道的问题，而且是众所周知的。但众所周知，一般的傅里叶级数，即使是函数 \(f(x)=1\)也不会收敛。在这种情况下，如果我们想让关于一般正交系统（ONS）\((\varphi_{n})\)的可微分函数具有收敛的傅里叶级数，我们必须找到系统\((\varphi_{n})\)的函数\(\varphi_{n}\)的特殊条件。本文对这一问题进行了研究。本文认为所得到的条件是最好的。本文考虑了一般正交系统的子系统。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文