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
Fejér and Wiener as also the extensions of theorems of Wiener given by
Lozinskiǐ, Keogh, Petersen and Matveev. Applications are made to the study of
coefficient properties of holomorphic functions in the unit disk with positive real
part.
