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 Mat. Zametki: Year: Volume: Issue: Page: Find

 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

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}