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If a complex power series $$f(z)=\sum_{n=0}^{\infty } a_n z^n $$ is uniformly convergent in any closed subset of the open disc $\mathrm{B} (0,R)$, and converges pointwise on $\partial\mathrm{B} $, is it uniformly convergent on $\overline{\mathrm{B} (0,R)} $?

I have tried to prove but have difficulties at some step, so I think of it as wrong. If it is wrong, are there some counterexamples?

TheWildCat
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1 Answers1

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On the boundary of the disc of convergence, the power series essentially reduces to a Fourier series. There are Fourier series that converge pointwise but not uniformly. You can find an example here: Pointwise but not uniform convergence of a Fourier series

One of the comments there also references Zygmund (Trigonometric Series I, p. 300). Zygmund mentions a similar result on power series following from the Fourier series one.

Kusma
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  • That example you cite is not "of power series type"; that is, not of the form $\sum_{n=0}^\infty c_ne^{int}$. – David C. Ullrich May 02 '18 at 14:47
  • Zygmund rewrites $f_{n,N}$ (he says he is following Fejer's proof) as a pure cosine polynomial $\sum_{k=N-n}^{N+n} a_k \cos(kx)$, which you can then turn into $\sum a_k e^{ikx}$, which is of power series type. – Kusma May 02 '18 at 15:50