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


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.

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