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Difference between revisions of "Kolmogorov-Seliverstov theorem"

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If the condition
 
If the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557301.png" /></td> </tr></table>
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$$\sum_{n=1}^\infty(a_n^2+b_n^2)W(n)<\infty$$
  
holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557302.png" />, then the Fourier series
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holds with $W(n)=\log n$, then the Fourier series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557303.png" /></td> </tr></table>
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$$\frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557304.png" /> can be taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557305.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557306.png" />, and this statement was strengthened in [[#References|[2]]], where its validity was proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557307.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557308.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k0557309.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055730/k05573010.png" /> a monotone majorant of the [[Lebesgue function|Lebesgue function]] on this set.
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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====
 
====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>
 
<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>

Latest revision as of 09:07, 27 July 2014

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 [1], [2]). In [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 [2], where its validity was proved for $\delta=0$ as well. This strong form was also obtained by A.I. Plessner [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 was proved, according to which one can take $W(n)=1$. S. Kaczmarz [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 on this set.

References

[1] A.N. Kolmogorov, G.A. Seliverstov, "Sur la convergence des séries de Fourier" C.R. Acad. Sci. Paris , 178 (1924) pp. 303–306
[2] A.N. Kolmogorov, G.A. Seliverstov, "Sur la convergence des séries de Fourier" Atti Accad. Naz. Lincei , 3 (1926) pp. 307–310
[3] A.I. Plessner, "Ueber Konvergenz von trigonometrischen Reihen" J. Reine Angew. Math. , 155 (1925) pp. 15–25
[4] S. Kaczmarz, "Sur la convergence et la sommabilité des développements orthogonaux" Studia Math. , 1 : 1 (1929) pp. 87–121
How to Cite This Entry:
Kolmogorov-Seliverstov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov-Seliverstov_theorem&oldid=18460
This article was adapted from an original article by S.A. Telyakovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article