On the order of partial sums of general orthogonal series

R. S. Davtyan

Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 451-462.

Résumé

It is shown that for convergence of every orthonormal system $\{\varphi_n(s)\}$ given on $[0,1]$, it is necessary and sufficient that, under the condition $\int_0^\infty\frac1{W^2(x)}dx+\infty$ on tlie increasing function $W(x)$ and for $\sum_{n=1}^\infty a_n^2=+\infty$ there hold $\left|\sum_{k=1}^na_k\varphi_k(x)\right|=o(W(\sum_1^ka_k^2))$ almost everywhere on $[0,1]$.