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Difference between revisions of "Lebesgue summation method"

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A method for summing [[Trigonometric series|trigonometric series]]. The series
 
A method for summing [[Trigonometric series|trigonometric series]]. 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/l/l057/l057940/l0579401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\tag{*}$$
  
is summable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l0579402.png" /> by the Lebesgue summation method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l0579403.png" /> if in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l0579404.png" /> of this point the integrated series
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is summable at a point $x_0$ by the Lebesgue summation method to the sum $s$ if in some neighbourhood $(x_0-h,x_0+h)$ of this point the integrated 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/l/l057/l057940/l0579405.png" /></td> </tr></table>
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$$\frac{a_0x}{2}+\sum_{n=1}^\infty\frac1n(a_n\sin nx-b_n\cos nx)$$
  
converges and its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l0579406.png" /> has symmetric derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l0579407.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l0579408.png" />:
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converges and its sum $F(x)$ has symmetric derivative at $x_0$ equal to $s$:
  
<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/l/l057/l057940/l0579409.png" /></td> </tr></table>
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$$\lim_{h\to0}\frac{F(x_0+h)-F(x_0-h)}{2h}=s.$$
  
 
The last condition can also be represented in the form
 
The last condition can also be represented in the form
  
<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/l/l057/l057940/l05794010.png" /></td> </tr></table>
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$$\lim_{h\to0}\left[\frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx_0+b_n\sin nx_0)\frac{\sin nh}{nh}\right]=s.$$
  
The Lebesgue summation method is not regular, in the sense that it is not possible to sum every convergent trigonometric series (*) (see [[Regular summation methods|Regular summation methods]]), but if (*) is the [[Fourier series|Fourier series]] of a summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l05794011.png" />, then it is summable almost-everywhere to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057940/l05794012.png" /> by the Lebesgue summation method. The method was proposed by H. Lebesgue [[#References|[1]]].
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The Lebesgue summation method is not regular, in the sense that it is not possible to sum every convergent trigonometric series \ref{*} (see [[Regular summation methods|Regular summation methods]]), but if \ref{*} is the [[Fourier series|Fourier series]] of a summable function $f$, then it is summable almost-everywhere to $f(x)$ by the Lebesgue summation method. The method was proposed by H. Lebesgue [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Leçons sur les séries trigonométriques" , Gauthier-Villars  (1906)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Leçons sur les séries trigonométriques" , Gauthier-Villars  (1906)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>

Revision as of 13:38, 27 September 2014

A method for summing trigonometric series. The series

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

is summable at a point $x_0$ by the Lebesgue summation method to the sum $s$ if in some neighbourhood $(x_0-h,x_0+h)$ of this point the integrated series

$$\frac{a_0x}{2}+\sum_{n=1}^\infty\frac1n(a_n\sin nx-b_n\cos nx)$$

converges and its sum $F(x)$ has symmetric derivative at $x_0$ equal to $s$:

$$\lim_{h\to0}\frac{F(x_0+h)-F(x_0-h)}{2h}=s.$$

The last condition can also be represented in the form

$$\lim_{h\to0}\left[\frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx_0+b_n\sin nx_0)\frac{\sin nh}{nh}\right]=s.$$

The Lebesgue summation method is not regular, in the sense that it is not possible to sum every convergent trigonometric series \ref{*} (see Regular summation methods), but if \ref{*} is the Fourier series of a summable function $f$, then it is summable almost-everywhere to $f(x)$ by the Lebesgue summation method. The method was proposed by H. Lebesgue [1].

References

[1] H. Lebesgue, "Leçons sur les séries trigonométriques" , Gauthier-Villars (1906)
[2] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
How to Cite This Entry:
Lebesgue summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_summation_method&oldid=33421
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article