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On Bary–Stechkin theorem
A. I. Rubinshtein National Research Nuclear University MEPhI, Kashirskoe road, 31, 115409, Moscow, Russia
Abstract:
In the beginning of the past century, N.N. Luzin proved almost everywhere convergence of an improper integral representing the function $\bar f$ conjugated to a $2\pi$-periodic summable with a square function $f(x)$. A few years later I.I. Privalov proved a similar fact for a summable function. V.I. Smirnov showed that if $\bar f$ is summable, then its Fourier series is conjugate to the Fourier series for $f(x)$. It is easy to see that if $f(x)\in\mathrm{Lip}\,\alpha$, $0<\alpha<1$, then $\bar f(x)\in\mathrm{Lip}\,\alpha$. The Hilbert transformation for $f(x)$ differs from $\bar f(x)$ by a bounded function and has a simpler kernel. It is easy to show that the Hilbert transformation of $f(x)\in\mathrm{Lip}\,\alpha$, $0<\alpha<1$, also belongs to $\mathrm{Lip}\,\alpha$. In 1956 N.K. Bari and S.B. Stechkin found the necessary and sufficient condition on the modulus of continuity $f(x)$ for the function $\bar f(x)$ to have the same modulus of continuity. In 2016, the author introduced the concept of conjugate function as Hilbert transformation for functions defined on a dyadic group. In the present paper we show an analogue of the Bari–Stechkin (and Privalov) theorem fails that for a conjugated in this sense function.
Keywords:
dyadic group, conjugate function, modulus of continuity, Bari–Stechkin theorem.
Received: 18.08.2017
Citation:
A. I. Rubinshtein, “On Bary–Stechkin theorem”, Ufimsk. Mat. Zh., 11:1 (2019), 68–71; Ufa Math. J., 11:1 (2019), 70–74
Linking options:
https://www.mathnet.ru/eng/ufa461https://doi.org/10.13108/2019-11-1-70 https://www.mathnet.ru/eng/ufa/v11/i1/p68
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Abstract page: | 306 | Russian version PDF: | 91 | English version PDF: | 28 | References: | 38 |
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