Men'shov-Rademacher theorem

A theorem on the almost-everywhere convergence of orthogonal series: If a system of functions $\{\phi_n(t)\}_{n=1}^\infty$ is orthonormal on a segment $[a,b]$ and if

$$\sum_{n=1}^\infty a_n^2\log^2n<\infty,$$

then the series

$$\sum_{n=1}^\infty a_n\phi_n(t)\label{*}\tag{*}$$

converges almost-everywhere on $[a,b]$. This result has been proved independently by D.E. Men'shov [1] and H. Rademacher [2]. Men'shov showed also that this assertion is sharp in the following sense. If a monotone increasing sequence $\omega(n)$ satisfies the condition $\omega(n)=o(\log^2n)$, then one can find an orthogonal series \eqref{*}, diverging everywhere, the coefficients of which satisfy the condition

$$\sum_{n=1}^\infty a_n^2\omega(n)<\infty.$$

References

[1]	D.E. Men'shov, "Sur la séries de fonctions orthogonales (I)" Fund. Math. , 4 (1923) pp. 82–105
[2]	H. Rademacher, "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen" Math. Ann. , 87 (1922) pp. 112–138
[3]	G. Alexits, "Konvergenzprobleme der Orthogonalreihen" , Deutsch. Verlag Wissenschaft. (1960)

References

[a1]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)