Concerning Fatou's theorem (Q1190055)

A sequence \(\{S_ n\}\) of complex numbers is almost convergent to \(L\) provided \(\lim_{m\to+\infty}(S_ p+\cdots+S_{p+m})/(m+1)=L\) uniformly with respect to \(p\). Let \(f(z)=\sum^ \infty_{n=0}a_ nz^ n\) for \(| z|<1\) and \(S_ n=a_ 0+a_ 1+\cdots+a_ n\). The author proves that if \(\{a_ n\}\) is almost convergent to 0 and \(f(z)\) is regular at \(z=1\), then \(\{S_ n\}\) is almost convergent to \(f(1)\). The author points out that this result is similar to a version of the theorem of Fatou.