On some submethod summability factors of infinite series

Bor proved a theorem dealing with the absolute Riesz summability of infinite series by using almost increasing sequences. In this study, a new absolute summability method is defined by means of the Riesz submethod, and within this framework, an extended version of Bor’s result on absolute Riesz summability factors is obtained under more general conditions. The theoretical results are then applied to Fourier series generated by orthogonal systems, and new sufficient conditions are established to guarantee their absolute summability by the proposed method. Thus, this study both extends existing results in the literature and provides further applications of absolute summability theory to Fourier analysis.