Infinite matrices summing every almost periodic sequence

Abstract

Necessary and sufficient conditions are given for infinite matrices to sum every almost periodic sequence and their basic properties as summability matrices are studied. It is then shown that these matrices enter naturally in the problem of the determination of the jump or total quadratic jump of normalized functions of bounded variation on the circle in terms of the limits of matrix transforms of certain functions of their Fourier-Stieltjes coefficients. The results obtained generalize the classical theorems of Fejr and Wiener as also the extensions of theorems of Wiener given by Lozinski, Keogh, Petersen and Matveev. Applications are made to the study of coefficient properties of holomorphic functions in the unit disk with positive real part.