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Difference between revisions of "Convergence multipliers"

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''for a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261201.png" /> of functions''
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{{TEX|done}}
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''for a series $\sum_{n=0}^\infty u_n(x)$ of functions''
  
Numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261203.png" /> such that the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261204.png" /> converges almost-everywhere on a measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261205.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261206.png" /> are numerical functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261207.png" />.
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Numbers $\lambda_n$, $n=0,1,\ldots,$ such that the series $\sum_{n=0}^\infty\lambda_nu_n(x)$ converges almost-everywhere on a measurable set $X$, where the $u_n(x)$ are numerical functions defined on $X$.
  
For example, for the trigonometric Fourier series of a function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261208.png" />, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c0261209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c02612010.png" /> are convergence multipliers (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c02612011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c02612012.png" /> can be chosen arbitrarily), i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c02612013.png" /> and if
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For example, for the trigonometric Fourier series of a function from $L_1$, the numbers $\lambda_n=1/\ln n$, $n=2,3,\ldots,$ are convergence multipliers ($\lambda_0$ and $\lambda_1$ can be chosen arbitrarily), i.e. if $f\in L_1[-\pi,\pi]$ 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/c/c026/c026120/c02612014.png" /></td> </tr></table>
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$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$
  
 
is its trigonometric Fourier series, then the series
 
is its trigonometric Fourier 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/c/c026/c026120/c02612015.png" /></td> </tr></table>
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$$\sum_{n=2}^\infty\frac{a_n\cos nx+b_n\sin nx}{\ln n}$$
  
converges almost-everywhere on the whole real line. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c02612016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026120/c02612017.png" />, then its trigonometric Fourier series itself converges almost-everywhere (see [[Carleson theorem|Carleson theorem]]).
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converges almost-everywhere on the whole real line. If $f\in L_p[-\pi,\pi]$, $p>1$, then its trigonometric Fourier series itself converges almost-everywhere (see [[Carleson theorem|Carleson theorem]]).

Latest revision as of 16:01, 17 July 2014

for a series $\sum_{n=0}^\infty u_n(x)$ of functions

Numbers $\lambda_n$, $n=0,1,\ldots,$ such that the series $\sum_{n=0}^\infty\lambda_nu_n(x)$ converges almost-everywhere on a measurable set $X$, where the $u_n(x)$ are numerical functions defined on $X$.

For example, for the trigonometric Fourier series of a function from $L_1$, the numbers $\lambda_n=1/\ln n$, $n=2,3,\ldots,$ are convergence multipliers ($\lambda_0$ and $\lambda_1$ can be chosen arbitrarily), i.e. if $f\in L_1[-\pi,\pi]$ and if

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

is its trigonometric Fourier series, then the series

$$\sum_{n=2}^\infty\frac{a_n\cos nx+b_n\sin nx}{\ln n}$$

converges almost-everywhere on the whole real line. If $f\in L_p[-\pi,\pi]$, $p>1$, then its trigonometric Fourier series itself converges almost-everywhere (see Carleson theorem).

How to Cite This Entry:
Convergence multipliers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_multipliers&oldid=32486
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article