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This article is cited in 14 scientific papers (total in 14 papers)
Convergence of Fourier series almost everywhere and in the $L$-metric
Sh. V. Kheladze
Abstract:
The following theorems are proved.
Theorem 1. There exists a constant $C>0$ such that for any function
$f\in L(0,2\pi)$ there is a measurable function $F$ for which $|F|=|f|$, and
a) $\displaystyle\int_0^{2\pi}\sup_n|S_n(F)(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$,
b) $\displaystyle\int_0^{2\pi}\sup_n|{\widetilde{S}}_n(F)(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$,
c) $\displaystyle\int_0^{2\pi}|\widetilde{F}(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$,
\noindent
where $S_n(F)$ is a partial sum of the Fourier series of $F$, $\widetilde S_n(F)$ is a partial sum of the conjugate Fourier series, and $\widetilde F$ is the conjugate function to $F$.
\medskip
Theorem 2. {\it For any function $f\in L(0,2\pi)$ and $\varepsilon>0$
there exists a measurable function $F$ such that $|F|=|f|$,
$\mu\{x\in[0,2\pi):F(x)\ne f(x)\}<\varepsilon$ ($\mu$ is Lebesgue measure), and both the Fourier series of $F$ and its conjugate series converge almost everywhere and in the metric of $L$.}
Bibliography: 11 titles.
Received: 20.12.1977
Citation:
Sh. V. Kheladze, “Convergence of Fourier series almost everywhere and in the $L$-metric”, Mat. Sb. (N.S.), 107(149):2(10) (1978), 245–258; Math. USSR-Sb., 35:4 (1979), 527–539
Linking options:
https://www.mathnet.ru/eng/sm2615https://doi.org/10.1070/SM1979v035n04ABEH001570 https://www.mathnet.ru/eng/sm/v149/i2/p245
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Abstract page: | 630 | Russian version PDF: | 130 | English version PDF: | 20 | References: | 81 |
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