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This article is cited in 1 scientific paper (total in 1 paper)
A description of the sets of Lebesque points and points of summability for a Fourier series
A. M. Diyachkov
Abstract:
The set of Lebesgue points of a locally integrable function on $N$-dimensional Euclidean space $\mathbf R^N$, $N\geqslant1$, is an $F_{\sigma\delta}$-set of full measure. In this article it is shown that every $F_{\sigma\delta}$-set of full measure is the set of Lebesgue points of some measurable bounded function, and, further, that a set with these properties is the set of points of convergence and nontangential (stable) convergence of a singular integral of convolution type:
$$
\varphi_\varepsilon\ast f(x), \quad \varphi_\varepsilon(t)=\varepsilon^{-N}\varphi(t/\varepsilon)\in L(\mathbf R^N), \quad \varepsilon\to+0,
$$
for some measurable bounded function $f$. On the basis of this result the set of points of summability of a multiple Fourier series by methods of Abel, Riesz, and Picard types is described.
Received: 01.06.1990
Citation:
A. M. Diyachkov, “A description of the sets of Lebesque points and points of summability for a Fourier series”, Math. USSR-Sb., 74:1 (1993), 111–118
Linking options:
https://www.mathnet.ru/eng/sm1373https://doi.org/10.1070/SM1993v074n01ABEH003338 https://www.mathnet.ru/eng/sm/v182/i9/p1367
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Abstract page: | 351 | Russian version PDF: | 132 | English version PDF: | 19 | References: | 46 | First page: | 1 |
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