# Abel-Poisson summation method

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)