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Mat. Zametki, 1976, Volume 19, Issue 4, Pages 481–490 (Mi mz7766)  

The summability of a special series by the $(C,\alpha)$ method

S. S. Agayan

Computing Center, Academy of Sciences of the Armenian SSR

Abstract: In the paper we study the problem of the summability by the $(C,\alpha)$ method of the special series
$$ f(x)\sim\sum_{n=-\infty}^{n=+\infty}c_n(x)\exp(in\mu(x)),\eqno(*) $$
where
\begin{gather*} c_n(x)=\frac2\pi\int_Gf(t)\exp(-in\mu(t))\frac{\sin1/2[\mu(t)-\mu(x)]}{t-x} dt,
\mu(x)=\frac1\pi\int_E\frac{dt}{t-x}. \end{gather*}
$E$ is some compactum on the real axis $R$ with positive Lebesgue measure and $G$ is the complement of $E$ with respect to $R$. It is shown that if the function $|f(t)|(1+|t|)^{-1}$ is integrable on $G$, then the series (*) is $(C,\alpha)$ summable at each Lebesgue point of the considered function $f$ and for any $\alpha>0$ coincides almost everywhere with $f(x)$.

Full text: PDF file (526 kB)

English version:
Mathematical Notes, 1976, 19:4, 295–300

Bibliographic databases:

UDC: 517.5
Received: 11.09.1975

Citation: S. S. Agayan, “The summability of a special series by the $(C,\alpha)$ method”, Mat. Zametki, 19:4 (1976), 481–490; Math. Notes, 19:4 (1976), 295–300

Citation in format AMSBIB
\Bibitem{Aga76}
\by S.~S.~Agayan
\paper The summability of a special series by the $(C,\alpha)$ method
\jour Mat. Zametki
\yr 1976
\vol 19
\issue 4
\pages 481--490
\mathnet{http://mi.mathnet.ru/mz7766}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=427895}
\zmath{https://zbmath.org/?q=an:0341.42009|0334.42012}
\transl
\jour Math. Notes
\yr 1976
\vol 19
\issue 4
\pages 295--300
\crossref{https://doi.org/10.1007/BF01156786}


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