Oct 172012
Let \(f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)\) be a power series. The radius of convergence of \(f\) is \(1\), and \(f\) is convergent at every point of the unit circle. If \(f(z)=0\) for every \(|z|=1\), then
\[a_n=0\]
for all nonnegative integer \(n\).
It seems that this is a particular case of an old Theorem from Cantor (1870), called
Cantor’s uniqueness theorem. If, for every real \(x\),
\[\lim_{N \rightarrow \infty} \sum_{n=-N}^N c_n e^{inx}=0,\]
then all the complex numbers \(c_n(n\in\Bbb Z)\) are zero.