The variation of general Fourier coefficients

Abstract

In this paper, series of modulus of differences of general Fourier coefficients are studied. This way we define the bounded variation of general Fourier coefficients of the function from some functional class. Throughout the paper we show that the sequence of general Fourier coefficients even of the function $g(x)=1$ is not of bounded variation. We find a condition on the functions of an orthonormal system (ONS), under which the sequence of Fourier coefficients of functions is of bounded variation, when the function is from the bounded variation class. We also study the behaviour of Fourier coefficients with respect to the classical ONS (trigonometric [1, Ch.1, §2], Haar [1, Ch. 1], Walsh [2] systems). In addition, we investigate the subsequences of ONS.