Abstract

On the basis of interpolation theory, several new inequalities are established for both general orthonormal systems and various specific classes of orthonormal systems including the Haar and Franklin systems and wavelets. The solution of the problem of characterizing the Fourier coefficients of continuous functions for general orthonormal systems is completed. For every complete orthonormal system, a continuous function is constructed that generates a universal singularity similar to the one appearing in Carleman’s theorem. This result significantly strengthens Olevskii’s theorem and turns into Orlicz’s theorem at the other end of the power scale. It is proved that the results obtained are, in a sense, final.

中文翻译：

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正交级数的不等式和完整正交系统的Carleman-Olevskii定理的加强

摘要

根据插值理论，为一般正交系统和各种特定类别的正交系统（包括Haar和Franklin系统以及小波）建立了几个新的不等式。完成了一般正交系统的连续函数傅立叶系数特征化问题的解决方案。对于每个完整的正交系统，都构造了一个连续函数，该函数产生与Carleman定理中出现的相似的通用奇点。这个结果大大增强了Olevskii定理，并在幂阶的另一端变成了Orlicz定理。从某种意义上说，证明了所得到的结果是最终的。