On the Fourier coefficients of functions of bounded variation

We notice that the Fourier coefficients of functions with bounded variation $V(0,1)$ with respect to general orthonormal systems (GONS) have no definite order of vanishing. In this connection we study the problem: which requirements to the GONS ensure that the Fourier coefficients of a function from $V(0,1)$ satisfy the inequality which holds for classical systems (trigonometric, Walsh, or Haar ones). In the present paper we study this problem and related issues.