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This article is cited in 1 scientific paper (total in 1 paper)
A criterion for the almost-everywhere convergence of Fourier–Walsh square partial sums of integrable functions
S. F. Lukomskii Pedagogical Institute of Saratov State University
Abstract:
S. V. Konyagin showed that if the one-dimensional Lebesgue constants $L_{n_k}$ for the Walsh–Paley system are unbounded, then the square partial sums $S_{n_k,n_k}(f)$ of some integrable function $f({x})=f(x_1,x_2)$ diverge almost everywhere. On the other hand the author constructed an example of sequence $\{n_k\}$ for which, sup $\sup L_{n_k}$ is finite, but for some integrable function $f({x})=f(x_1,x_2)$ the partial sums $S_{n_k,n_k}(f)$ diverge almost everywhere. Thus boundedness of the Lebesgue constants $L_{n_k}$ is not a necessary and sufficient condition for the convergence almost everywhere of the partial sums $S_{n_k,n_k}(f)$ of any integrable function. In this article we find such a necessary and sufficient condition.
Received: 16.06.1994
Citation:
S. F. Lukomskii, “A criterion for the almost-everywhere convergence of Fourier–Walsh square partial sums of integrable functions”, Mat. Sb., 186:7 (1995), 133–146; Sb. Math., 186:7 (1995), 1057–1070
Linking options:
https://www.mathnet.ru/eng/sm56https://doi.org/10.1070/SM1995v186n07ABEH000056 https://www.mathnet.ru/eng/sm/v186/i7/p133
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Abstract page: | 295 | Russian version PDF: | 104 | English version PDF: | 6 | References: | 35 | First page: | 1 |
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