Parseval's equality for Fourier-Stieltjes series (Q1405916)

The main result of the paper is contained in the following assertion. Let the function \(g\) have a bounded variation, the \(2\pi\)-periodic function \(f\) be piece-wise continuous and the functions \(f\) and \(g\) do not have common separation points. Then for the functions \(f\) and \(g\) \[ \frac{1}{\pi} \int\limits_{-\pi}^\pi f(x) dg(x) = \frac 12 a_0\alpha_0 + \sum\limits_{k=1}^\infty (a_k\alpha_k + b_k\beta_k),\tag{1} \] where \(a_n\) and \(b_n\) are the coefficients of the Fourier series of the function \(f\), \( \alpha_k\) and \(\beta_k\) are the coefficients of the Fourier-Stieltjes series for the function \(g\) with respect to cosines and sines, respectively.