Some Problems of Convergence of General Fourier Series

Pub Date : 2022-12-23 DOI:10.3103/s1068362322060085

V. Tsagareishvili, G. Tutberidze

引用次数: 0

Abstract

Banach [1] proved that good differential properties of function do not guarantee the a.e. convergence of the Fourier series of this function with respect to general orthonormal systems (ONS). On the other hand it is very well known that a sufficient condition for the a.e. convergence of an orthonormal series is given by the Menshov–Rademacher Theorem. The paper deals with sequence of positive numbers \((d_{n})\) such that multiplying the Fourier coefficients \((C_{n}(f))\) of functions with bounded variation by these numbers one obtains a.e. convergent series of the form \(\sum_{n=1}^{\infty}d_{n}C_{n}(f)\varphi_{n}(x).\) It is established that the resulting conditions are best possible.

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一般傅里叶级数收敛性的几个问题

摘要banach[1]证明了函数的良好的微分性质并不能保证该函数的傅里叶级数对一般正交系统(ONS)的a.e.收敛。另一方面，众所周知，Menshov-Rademacher定理给出了一个标准正交级数a.e.收敛的充分条件。本文讨论了正数序列\((d_{n})\)，使有界变分函数的傅里叶系数\((C_{n}(f))\)与这些数相乘，得到了形式为\(\sum_{n=1}^{\infty}d_{n}C_{n}(f)\varphi_{n}(x).\)的收敛级数，并证明了所得到的条件是最优的。

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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