Difference between revisions of "AbelPoisson summation method"
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Revision as of 12:35, 10 January 2020
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
$$\lim_{\rho\to10}f(\rho,\phi)=S,$$
where
$$f(\rho,\phi)=\frac{a_0}{2}+\sum_{k=1}^\infty(a_k\cos k\phi+b_k\sin k\phi)\rho^k,$$
$$f(\rho,\phi)=\frac1\pi\int\limits_{\pi}^\pi f(\phi+t)\frac{1\rho^2}{2(12\rho\cos t+\rho^2)}dt.\tag{*}$$
If $f\in C(0,2\pi)$, then the integral on the righthand 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 $P(1,\phi_0)$ nontangentially inside the disc, cf. [a2], pp. 129130.
References
[a1]  A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) 
[a2]  M. Tsuji, "Potential theory in modern function theory" , Maruzen (1975) 
AbelPoisson summation method. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=AbelPoisson_summation_method&oldid=44330