A criterion for satisfying Parseval's equality with a bounded and a summable function (Q1317570)

The Parseval's equality \[ {1\over 2\pi} \int^{2\pi}_ 0 f(x) g(x) dx= {a_ 0 \alpha_ 0\over 2}+ \sum^ \infty_{n= 1} (a_ n \alpha_ n+ b_ n\beta_ n)\tag{P} \] (where \(a_ n\), \(b_ n\) and \(\alpha_ n\) and \(\beta_ n\) are the Fourier coefficients of \(f\) and \(g\), respectively) does not hold in general for \(f\) continuous \(2\pi\) periodic and \(g\) summable on \([0,2\pi]\). The author shows that (P) holds for the given bounded function \(f\) and for any \(g\in L[0,2\pi]\) if and only if the partial sums \(S_ n(f; x)\) of the Fourier series of \(f\) are bounded uniformly with respect to \(n\) and \(x\).