Abstract

The integrable function universal for classes \(L^{p}[0,1]\), \(p\in\left(0,1\right)\) with respect to the Walsh system is constructed in terms of signs. The Fourier–Walsh series of this function converges everywhere in \((0,1)\) and in the metrics \(L^{p}[0,1]\), \(p\in(0,1)\), and its Fourier coefficients by the Walsh system are in decreasing order. After selecting the appropriate signs for its Fourier coefficients, the newly obtained series will be universal with respect to permutations in \(L^{p}[0,1]\), \(p\in\left(0,1\right)\).

中文翻译：

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关于沃尔什系统的通用功能

摘要

关于Walsh系统的类\（L ^ {p} [0,1] \），\（p \ in \ left（0,1 \ right）\）的通用可积分函数是根据符号构造的。此函数的Fourier–Walsh级数在\（（0,1）\）和度量\（L ^ {p} [0,1] \），\（p \ in（0,1）\ ），而Walsh系统的傅立叶系数则按降序排列。在为其傅里叶系数选择适当的符号后，新获得的级数对于\（L ^ {p} [0,1] \），\（p \ in \ left（0,1 \ right ）\）。