The variational properties of general Fourier coefficients of functions with derivatives of finite variation

Abstract In the present paper, we consider problems of convergence of a special series of moduli of the difference of the general Fourier coefficients of functions with derivatives of finite variation with respect to general orthonormal systems (ONS). It is shown that such series with respect to the Haar system or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msqrt> <m:mn>2</m:mn> </m:msqrt> <m:mo>⁢</m:mo> <m:mrow> <m:mi>cos</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:mi>n</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\sqrt{2}\cos 2\pi nx)} system are convergent for functions with derivatives of finite variation. Furthermore, it is proved that, with respect to general ONS, this series, in general, does not converge. Also, we find the optimal condition that needs to be imposed on the functions of the ONS so that the general orthonormal systems, possessing the properties similar to the Haar system, converge for functions with derivatives of finite variation.