论一般偏和的有界性

Pub Date : 2023-12-24 DOI:10.1007/s10998-023-00565-y

Vakhtang Tsagareishvili

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引用次数: 0

摘要

从 S. Banach 的结果可以看出，即使是函数 \(f(x)=1\)\((x\in[0,1]))的一般傅里叶级数的一般部分和在[0,1]上也不是有界的。在本文中，我们为正交系统 \((\varphi _n\))的函数 \((\varphi _n\))找到了条件，在这些条件下，来自某个可微分类的函数的偏和是有界的。我们证明所得到的结果是最好的。我们还研究了一般正交系统子序列的性质。

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On the boundedness of general partial sums

From S. Banach’s results it follows that even for the function \(f(x)=1\) \((x\in [0,1])\) the general partial sums of its general Fourier series are not bounded a.e. on [0, 1]. In the present paper, we find conditions for the functions \(\varphi _n\) of an orthonormal system \((\varphi _n\)) under which the partial sums of functions from some differentiable class are bounded. We prove that the obtained results are best possible. We also investigate the properties of subsequences of general orthonormal systems.

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