The summability of a special series by the (C, α) method

Abstract

In the paper we study the problem of the summability by the (C,α) method of the special series

$$f(x) \sim \sum\nolimits_{n = - \infty }^{n = + \infty } {c_n (x)\exp (in^\mu } (x)),$$

((*))

where

$$\begin{gathered} c_n (x) = \frac{2}{\pi }\mathop \smallint \nolimits_G f(t)\exp ( - in^\mu (t))\frac{{\sin {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}[\mu (t) - \mu (x)]}}{{t - x}}dt, \hfill \\ \mu (x) = \frac{1}{\pi }\mathop \smallint \nolimits_E \frac{{dt}}{{t - x}}, \hfill \\ \end{gathered}$$

. 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,α) summable at each Lebesgue point of the considered function f and for anyα > 0 coincides almost everywhere with f(x).