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Difference between revisions of "Abel-Poisson summation method"

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One of the methods for summing Fourier series. The Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a0101501.png" /> is summable by the Abel–Poisson method at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a0101502.png" /> to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a0101503.png" /> if
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{{TEX|done}}
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One of the methods for summing Fourier series. The Fourier series of a function $f\in L[0,2\pi]$ is summable by the Abel–Poisson method at a point $\phi$ to a number $S$ 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/a/a010/a010150/a0101504.png" /></td> </tr></table>
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$$\lim_{\rho\to1-0}f(\rho,\phi)=S,$$
  
 
where
 
where
  
<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/a/a010/a010150/a0101505.png" /></td> </tr></table>
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$$f(\rho,\phi)=\frac{a_0}{2}+\sum_{k=1}^\infty(a_k\cos k\phi+b_k\sin k\phi)\rho^k,$$
  
<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/a/a010/a010150/a0101506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$f(\rho,\phi)=\frac1\pi\int\limits_{-\pi}^\pi f(\phi+t)\frac{1-\rho^2}{2(1-2\rho\cos t+\rho^2)}dt.\tag{*}$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a0101507.png" />, then the integral on the right-hand side is a harmonic function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a0101508.png" />, which is, as has been shown by S. Poisson, a solution of the Dirichlet problem for the disc. The [[Abel summation method|Abel summation method]] applied to Fourier series was therefore named the Abel–Poisson summation method, and the integral (*) was named the [[Poisson integral|Poisson integral]].
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If $f\in C(0,2\pi)$, then the integral on the right-hand side is a harmonic function for $|z|\equiv|\rho e^{i\phi}|<1$, which is, as has been shown by S. Poisson, a solution of the Dirichlet problem for the disc. The [[Abel summation method|Abel summation method]] applied to Fourier series was therefore named the Abel–Poisson summation method, and the integral \ref{*} was named the [[Poisson integral|Poisson integral]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a0101509.png" /> are polar coordinates of a point inside the disc of radius one, then one can consider the limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015010.png" /> as the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015011.png" /> approaches a point on the bounding circle not by a radial or by a tangential but rather along an arbitrary path. In this situation the Schwarz theorem applies: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015012.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015013.png" /> and is continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015014.png" />, then
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If $(\rho,\phi)$ are polar coordinates of a point inside the disc of radius one, then one can consider the limit of $f(\rho,\phi)$ as the point $M(\rho,\phi)$ approaches a point on the bounding circle not by a radial or by a tangential but rather along an arbitrary path. In this situation the Schwarz theorem applies: If $f$ belongs to $L[0,2\pi]$ and is continuous at a point $\phi_0$, then
  
<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/a/a010/a010150/a01015015.png" /></td> </tr></table>
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$$\lim_{(\rho,\phi)\to(1,\phi_0)}f(\rho,\phi)=f(\phi_0)$$
  
irrespective of the path along which the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015016.png" /> approaches the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015017.png" /> as long as that path remains within the disc with radius one.
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irrespective of the path along which the point $M(\rho,\phi)$ approaches the point $P(1,\phi_0)$ as long as that path remains within the disc with radius one.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
A theorem related to Schwarz' theorem stated above is Fatou's theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015018.png" />, then for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015019.png" />
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A theorem related to Schwarz' theorem stated above is Fatou's theorem: If $f\in L[0,2\pi]$, then for almost all $\phi_0$
  
<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/a/a010/a010150/a01015020.png" /></td> </tr></table>
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$$\lim_{(\rho,\phi)\to(1,\phi_0)}f(\rho,\phi)=f(\phi_0)$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015021.png" /> approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010150/a01015022.png" /> non-tangentially inside the disc, cf. [[#References|[a2]]], pp. 129-130.
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as $M(\rho,\phi)$ approaches $P(1,\phi_0)$ non-tangentially inside the disc, cf. [[#References|[a2]]], pp. 129-130.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Tsuji,  "Potential theory in modern function theory" , Maruzen  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Tsuji,  "Potential theory in modern function theory" , Maruzen  (1975)</TD></TR></table>

Revision as of 11:44, 2 September 2014

One of the methods for summing Fourier series. The Fourier series of a function $f\in L[0,2\pi]$ is summable by the Abel–Poisson method at a point $\phi$ to a number $S$ if

$$\lim_{\rho\to1-0}f(\rho,\phi)=S,$$

where

$$f(\rho,\phi)=\frac{a_0}{2}+\sum_{k=1}^\infty(a_k\cos k\phi+b_k\sin k\phi)\rho^k,$$

$$f(\rho,\phi)=\frac1\pi\int\limits_{-\pi}^\pi f(\phi+t)\frac{1-\rho^2}{2(1-2\rho\cos t+\rho^2)}dt.\tag{*}$$

If $f\in C(0,2\pi)$, then the integral on the right-hand side is a harmonic function for $|z|\equiv|\rho e^{i\phi}|<1$, which is, as has been shown by S. Poisson, a solution of the Dirichlet problem for the disc. The Abel summation method applied to Fourier series was therefore named the Abel–Poisson summation method, and the integral \ref{*} was named the Poisson integral.

If $(\rho,\phi)$ are polar coordinates of a point inside the disc of radius one, then one can consider the limit of $f(\rho,\phi)$ as the point $M(\rho,\phi)$ approaches a point on the bounding circle not by a radial or by a tangential but rather along an arbitrary path. In this situation the Schwarz theorem applies: If $f$ belongs to $L[0,2\pi]$ and is continuous at a point $\phi_0$, then

$$\lim_{(\rho,\phi)\to(1,\phi_0)}f(\rho,\phi)=f(\phi_0)$$

irrespective of the path along which the point $M(\rho,\phi)$ approaches the point $P(1,\phi_0)$ as long as that path remains within the disc with radius one.

References

[1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)


Comments

A theorem related to Schwarz' theorem stated above is Fatou's theorem: If $f\in L[0,2\pi]$, then for almost all $\phi_0$

$$\lim_{(\rho,\phi)\to(1,\phi_0)}f(\rho,\phi)=f(\phi_0)$$

as $M(\rho,\phi)$ approaches $P(1,\phi_0)$ non-tangentially inside the disc, cf. [a2], pp. 129-130.

References

[a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[a2] M. Tsuji, "Potential theory in modern function theory" , Maruzen (1975)
How to Cite This Entry:
Abel-Poisson summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel-Poisson_summation_method&oldid=18630
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article