Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be a power series, $a_n, z\in \mathbb{C}$. Suppose the radius of convergence of $f$ is $1$, and $f$ is convergent at every point of the unit circle.

Question:If $f(z)=0$ for every $|z|=1$, then can we draw the conclusion that $a_n=0$ for all nonnegative integer $n$?

I think the answer is yes, but I failed to prove it. My approach is concerning about the function $F_\lambda(z):=f(\lambda z)$ for $0\leq\lambda\leq 1$, $|z|=1$. Abel's theorem shows that $F_\lambda$ converge to $F_1$ pointwisely as $\lambda\rightarrow 1$ on the unit circle. If I have the property that $f$ is bounded in the unit disk, then I can apply Lebesgue's dominated convergence theorem to prove $a_0=0$, and by induction I can prove $a_n=0$ for all $n$. However, I cannot prove $f(z) $ is bounded in the unit disk.

Any answers or comments are welcome. I'll really appreciate your help.

  • $\begingroup$ Maybe I'm making some very stupid mistake, but isn't $f$ continuous on the unit disk since it's the pointwise limit of uniformly continuous functions? $\endgroup$
    – k.stm
    Oct 8 '12 at 11:27
  • $\begingroup$ Can you use the Maximum modulus theorem? $\endgroup$
    – PAD
    Oct 8 '12 at 11:37
  • 6
    $\begingroup$ A pointwise limit of uniformly continuous functions need not be continuous. $\endgroup$
    – GEdgar
    Oct 8 '12 at 12:55
  • $\begingroup$ @jerrysciencemath : Are you satisfied with my answer below? $\endgroup$ Oct 12 '12 at 20:37
  • $\begingroup$ @MalikYounsi: I'm sorry that I'm busy doing other things these days, I haven't check the document in your reply, but it seems to be a satisfactory answer~ If I have time, I will check it through details, thanks for your answer! $\endgroup$
    – Yuchen Liu
    Oct 13 '12 at 8:28

It seems to me that this is a particular case of an old Theorem from Cantor (1870), called Cantor's uniqueness theorem. The theorem says that if, for every real $x$, $$\lim_{N \rightarrow \infty} \sum_{n=-N}^N c_n e^{inx}=0,$$ then all the complex numbers $c_n$'s are zero.

You can google "Uniqueness of Representation by Trigonometric Series" for more information. See e.g. this document for a proof and some history of the result.

  • 1
    $\begingroup$ Yes, I think this correct: +1. By the way, you seem to know a lot about holomorphic functions: are they a field you specialize in ? $\endgroup$ Oct 8 '12 at 21:33
  • $\begingroup$ @GeorgesElencwajg : Thank you, I also think it is correct. This question really is more about trigonometric series than anything else : the fact that $f(x)$ converges on the unit circle implies that the radius of convergence is $\geq 1$, and if the radius is $>1$ then the result is trivial... So the hypothesis about the radius of convergence does not mean much. About holomorphic functions, I am a PhD student in complex analysis, so I've read a lot about this subject. I'm interested in other fields too though. $\endgroup$ Oct 9 '12 at 12:42
  • 1
    $\begingroup$ Thanks for your explanations, Malik, and best wishes for your PhD. $\endgroup$ Oct 9 '12 at 14:49
  • $\begingroup$ I'm removing an answer I gave which was way off. Thanks to those who pointed that out... $\endgroup$
    – coffeemath
    Oct 16 '12 at 2:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.