Abel-Poisson summation method
One of the methods for summing Fourier series. The Fourier series of a function is summable by the Abel–Poisson method at a point
to a number
if
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where
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![]() | (*) |
If , then the integral on the right-hand side is a harmonic function for
, 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 (*) was named the Poisson integral.
If are polar coordinates of a point inside the disc of radius one, then one can consider the limit of
as the point
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
belongs to
and is continuous at a point
, then
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irrespective of the path along which the point approaches the point
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 , then for almost all
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as approaches
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) |
Abel-Poisson summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel-Poisson_summation_method&oldid=18630