Article
Ufa Mathematical Journal
Volume 11, Number 1, pp. 70-74
On Bary -Stechkin theorem
Rubinshtein A.I.
DOI:10.13108/2019-11-1-70
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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 $f(x)\in\mathop{\mathrm{Lip}}\alpha$, $0<\alpha<1$, then $\bar f(x)\in\mathop{\mathrm{Lip}}\alpha$.
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 $f(x)\in\mathop{\mathrm{Lip}}\alpha$, $0<\alpha<1$, also belongs to $\mathop{\mathrm{Lip}}\alpha$.
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.