The absolute convergence of orthogonal series

A. S. Zinov'ev

Matematičeskie zametki, Tome 12 (1972) no. 5, pp. 511-516.

Résumé

We obtain sufficient conditions for the absolute convergence of Fourier series for functions of $\mathrm{L}^2_{\mathrm{d}\psi}$ depending on the properties of the function being expanded and the rate of growth of the sums $\sum_{k=1}^n\varphi_k^2(x)$ of the system of functions $\{\varphi_k(\mathrm{t})\}$ orthonormalized in $[a,\mathrm{ b}]$ with respect to $\mathrm{d}\psi(\mathrm{t})$. We show that if at some point $x\in[a,\mathrm{b}]$ the function $\psi(\mathrm{t})$ has a discontinuity, at that point the Fourier series of any function $f(\mathrm{t})\in \mathrm{L}_{\mathrm{d}\psi}^2$, converges absolutely.