View source for Kolmogorov-Seliverstov theorem

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If the condition

$$\sum_{n=1}^\infty(a_n^2+b_n^2)W(n)<\infty$$

holds with $W(n)=\log n$, then the Fourier series

$$\frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)$$

converges almost-everywhere. This was established by A.N. Kolmogorov and G.A. Seliverstov (see [[#References|[1]]], [[#References|[2]]]). In [[#References|[1]]] it was actually proved that $W(n)$ can be taken to be $\log^{1+\delta}n$ for any $\delta>0$, and this statement was strengthened in [[#References|[2]]], where its validity was proved for $\delta=0$ as well. This strong form was also obtained by A.I. Plessner [[#References|[3]]]. Prior to the Kolmogorov–Seliverstov theorem, the theorem (G.H. Hardy, 1916) was known with $W(n)=\log^2 n$. The Kolmogorov–Seliverstov theorem remained the strongest result in this direction until 1966, when the [[Carleson theorem|Carleson theorem]] was proved, according to which one can take $W(n)=1$. S. Kaczmarz [[#References|[4]]] transferred the Kolmogorov–Seliverstov theorem from the trigonometric system to arbitrary orthonormal systems by proving that for the almost-everywhere convergence of series in such systems on some set, one can take for $W(n)$ a monotone majorant of the [[Lebesgue function|Lebesgue function]] on this set.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   G.A. Seliverstov,   "Sur la convergence des séries de Fourier"  ''C.R. Acad. Sci. Paris'' , '''178'''  (1924)  pp. 303–306</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   G.A. Seliverstov,   "Sur la convergence des séries de Fourier"  ''Atti Accad. Naz. Lincei'' , '''3'''  (1926)  pp. 307–310</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Plessner,   "Ueber Konvergenz von trigonometrischen Reihen"  ''J. Reine Angew. Math.'' , '''155'''  (1925)  pp. 15–25</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kaczmarz,   "Sur la convergence et la sommabilité des développements orthogonaux"  ''Studia Math.'' , '''1''' :  1  (1929)  pp. 87–121</TD></TR></table>