Let
be Lebesgue Integrable and let
![\begin{displaymath}
f(r,\theta)={1\over 2\pi}\int_{-\pi}^\pi f(t){1-r^2\over 1-2r\cos(t-\theta)+r^2}\,dt
\end{displaymath}](f_340.gif) |
(1) |
be the corresponding Poisson Integral. Then Almost Everywhere in
![\begin{displaymath}
\lim_{r\to 0^-} f(r,\theta)=f(\theta).
\end{displaymath}](f_342.gif) |
(2) |
Let
![\begin{displaymath}
F(z)=c_0+c_1z+c_2z^2+\ldots+c_nz^n+\ldots
\end{displaymath}](f_343.gif) |
(3) |
be regular for
, and let the integral
![\begin{displaymath}
{1\over 2\pi}\int_{-\pi}^\pi \vert F(re^{i\theta})\vert^2\,d\theta
\end{displaymath}](f_345.gif) |
(4) |
be bounded for
. This condition is equivalent to the convergence of
![\begin{displaymath}
\vert c_0\vert^2+\vert c_1\vert^2+\ldots+\vert c_n\vert^2+\ldots.
\end{displaymath}](f_347.gif) |
(5) |
Then almost everywhere in
,
![\begin{displaymath}
\lim_{r\to 0^-} F(re^{i\theta})=F(e^{i\theta}).
\end{displaymath}](f_349.gif) |
(6) |
Furthermore,
is measurable,
is Lebesgue Integrable, and the Fourier
Series of
is given by writing
.
References
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 274, 1975.
© 1996-9 Eric W. Weisstein
1999-05-26