On the saturation class of the Abel’s method of summability of the series of Faber polynomials
In the introduction the object of investigation is pointed out - the series of Faber polynomials. The basic notions, such as saturation, degree of saturation, class of saturation are defined. The purpose of the research is the obtaining of structural characteristic of the saturation class of the Abel’s method of summability of the series of Faber polynomials. In the main part the structural characteristics of the saturation class are obtained: 1) in the Jordan domains with smooth boundary of S. Ya. Alper curve; 2) in the Jordan domains with rectifiable boundary curve of bounded rotation and free from cusps; 3) for segment [-1, 1]. In the Jordan domains with smooth boundary of S. Ya. Alper curve the saturation class is characterized in terms of the ordinary derivative. In the Jordan domains with the rectifiable boundary curve of bounded rotation and free from cusps the saturation class is characterized in terms of the Faber derivative. For segment [-1, 1] the saturation class is characterized in terms of the ordinary derivative of the trigonometric conjugate function to the composite function <i>f</i>(cos<i>x</i>). The obtained results can be applied in the theoretical research of summability of the series of Faber polynomials, as well as in special courses of approximation theory.