Abel-Poisson summation method

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)