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 Mat. Sb. (N.S.), 1973, Volume 91(133), Number 4(8), Pages 537–553 (Mi msb3321)

Integrability of trigonometric series. The estimation of the integral modulus of continuity

S. A. Telyakovskii

Abstract: Let $a_m$ tend to zero and let the quantities
\begin{align*} B_n&=\sum_{m=1}^n(\frac mn)^k|\Delta a_m|+\sum_{m=n+1}^\infty|\Delta a_m|+
&\qquad+\sum_{m=2}^n(\frac mn)^k|\sum_{i=1}^{[m/2]}\frac{\Delta a_{m-i}-\Delta a_{m+i}}i|+\sum_{m=n+1}^\infty|\sum_{i=1}^{[m/2]}\frac{\Delta a_{m-i}-\Delta a_{m+i}}i|. \end{align*}
be finite. We put $f(x)=\frac{a_0}2+\sum_{m=1}^\infty a_m\cos mx$ and $g(x)=\sum_{m=1}^\infty a_m\sin mx$.
It is shown that the integral modulus of continuity of $k$th order for the function $f$ satisfies the estimate $\omega_k(f,\frac1n)_L=O(B_n)$, and that if the series $\sum\frac{|a_m|}m$, converges then
$$\omega_k(g,\frac1n)_L=\frac{2^k}\pi\sum_{m=n}^\infty\frac{|a_m|}m+O(B_n).$$

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English version:
Mathematics of the USSR-Sbornik, 1973, 20:4, 557–573

Bibliographic databases:

UDC: 517.522.3
MSC: 26A15, 42A16

Citation: S. A. Telyakovskii, “Integrability of trigonometric series. The estimation of the integral modulus of continuity”, Mat. Sb. (N.S.), 91(133):4(8) (1973), 537–553; Math. USSR-Sb., 20:4 (1973), 557–573

Citation in format AMSBIB
\Bibitem{Tel73} \by S.~A.~Telyakovskii \paper Integrability of trigonometric series. The estimation of the integral modulus of continuity \jour Mat. Sb. (N.S.) \yr 1973 \vol 91(133) \issue 4(8) \pages 537--553 \mathnet{http://mi.mathnet.ru/msb3321} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=330888} \zmath{https://zbmath.org/?q=an:0279.42005} \transl \jour Math. USSR-Sb. \yr 1973 \vol 20 \issue 4 \pages 557--573 \crossref{https://doi.org/10.1070/SM1973v020n04ABEH001982}