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

where

(*)

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

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

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)