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

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 $f(x)$.