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Difference between revisions of "Men'shov-Rademacher theorem"

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A theorem on the almost-everywhere convergence of orthogonal series: If a system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063430/m0634301.png" /> is orthonormal on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063430/m0634302.png" /> and if
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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
  
<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/m/m063/m063430/m0634303.png" /></td> </tr></table>
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$$\sum_{n=1}^\infty a_n^2\log^2n<\infty,$$
  
 
then the series
 
then the 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/m/m063/m063430/m0634304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\sum_{n=1}^\infty a_n\phi_n(t)\tag{*}$$
  
converges almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063430/m0634305.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063430/m0634306.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063430/m0634307.png" />, then one can find an orthogonal series (*), diverging everywhere, the coefficients of which satisfy the condition
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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 \ref{*}, diverging everywhere, the coefficients of which satisfy 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/m/m063/m063430/m0634308.png" /></td> </tr></table>
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$$\sum_{n=1}^\infty a_n^2\omega(n)<\infty.$$
  
 
====References====
 
====References====

Revision as of 16:56, 16 August 2014

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

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

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 \ref{*}, 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)


Comments

References

[a1] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Men'shov-Rademacher theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Men%27shov-Rademacher_theorem&oldid=22807
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article