On the summability Legendre series by Abel method

Abel’s method is one of the well-known among linear methods of summability of series. Abel's method of summability for trigonometric Fourier series was first applied by Poisson, so this method is sometimes called the Abel- Poisson method. Subsequently, Fatou considered the summability of the series obtained by differentiating the Fourier trigonometric series. Gobson studied the summability of the Legendre series by Abel's method. In particular, he showed that if the function is continuous at a point, its Legendre series would be summability by Abel's method to the value of the function at this point, and in the case of discontinuity of the first kind at the point, to the arithmetic mean of the right and left limits of the function. Gobson also studied the conditions of uniform summability of Legendre's series by Abel's method. In this paper, the issue of almost everywhere summability of Legendre's series by Abel's method is considered. In particular, the so-called points D are determined and it is shown that at each point D the Legendre series is summability by Abel's method. In the paper summability of Legendre's derived series by Abel's methods is studed. The form of the Abelian mean of the derived series, some of its properties, and the sufficient condition for the existence of the limit of the derivative of the Abel mean have been determined.