Riemann-Stieltjes integrals and Parseval's equality (Q1425494)

It is proved that if \(2\pi\)-periodic function \(f(x)\) and a function \(G(x)\), which is a sum of \(2\pi\)-periodic and linear functions, are Lebesgue integrable and \(f(x)\) is integrable over a period (any segment of length \(2\pi\)) in the sense of Riemann-Stieltjes by \(\overline{G(x)}\), then the Parseval equality \[ \frac12\,(\mathcal R-\mathcal S)\int\limits_0^{2\pi} f(x)\,d\overline{G(x)} = (\mathcal R,2)\sum\limits_{k=-\infty}^{+\infty} \hat f(k)\, \overline{\widehat{dG}(k)} \] holds, where \(\hat f(k)\) and \(\widehat{dG}(k)\) are the Fourier coefficients of \(f(x)\) and the Fourier-Stieltjes coefficients of \(G(x)\), respectively, the integral in the equality is the Riemann-Stieltjes integral and the series in the right-side part of the equality may not converge but is summable by the Riemann method \((\mathcal R,2)\).