View source for Men'shov-Rademacher theorem

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR></table>