Fatou's Theorems

Let $f(\theta)$ be Lebesgue Integrable and let

$$f(r,\theta)={1\over 2\pi}\int_{-\pi}^\pi f(t){1-r^2\over 1-2r\cos(t-\theta)+r^2}\,dt$$ (1)

be the corresponding Poisson Integral. Then Almost Everywhere in $-\pi\leq\theta\leq\pi,$

$$\lim_{r\to 0^-} f(r,\theta)=f(\theta).$$ (2)

Let

$$F(z)=c_0+c_1z+c_2z^2+\ldots+c_nz^n+\ldots$$ (3)

be regular for $\vert z\vert<1$, and let the integral

$${1\over 2\pi}\int_{-\pi}^\pi \vert F(re^{i\theta})\vert^2\,d\theta$$ (4)

be bounded for $r<1$. This condition is equivalent to the convergence of

$$\vert c_0\vert^2+\vert c_1\vert^2+\ldots+\vert c_n\vert^2+\ldots.$$ (5)

Then almost everywhere in $-\pi\leq\theta\leq\pi$,

$$\lim_{r\to 0^-} F(re^{i\theta})=F(e^{i\theta}).$$ (6)

Furthermore, $F(e^{i\theta})$ is measurable, $\vert F(e^{i\theta})\vert^2$ is Lebesgue Integrable, and the Fourier Series of $F(e^{i\theta})$ is given by writing $z=e^{i\theta}$.

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 274, 1975.