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I am looking for a power series $\displaystyle f(z) = \sum_{n=0}^{+\infty} a_n z^n$ that converge uniformly on $\mathcal{D} = \Big\{ z \in \mathbb{C} \ / \ \vert z \vert \leq 1 \Big\}$ but not normally. Of course, a proof that this is impossible would be even better. It seems close to this question, but it's not quite the same.

It is well-known that normal convergence implies uniform convergence, and that the converse is false but I haven't found yet a counterexample in the form of a power series on $\mathcal{D}$.

In other words, I would like a complex sequence $(a_n)_{n \in \mathbb{N}}$ satisfying the three conditions :

  • $\displaystyle\sum_{n=0}^{+\infty} a_n z^n$ converges for all $z$ such that $\vert z \vert \leq 1$
  • $\displaystyle \sup_{\vert z \vert \leq 1} \left\vert \sum_{n=N}^{+\infty} a_n z^n \right\vert \longrightarrow 0$ when $N\to+\infty$
  • $\displaystyle\sum_{n=0}^{+\infty} \vert a_n \vert = +\infty$
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I had to look up "normal convergence": eom.springer.de/N/n067430.htm –  Yemon Choi Jan 25 '10 at 22:31
    
Me too. Fortunately, as indicated at the end of the question, in this case it is the same as absolute convergence. (More generally it is the stronger condition that the hypotheses of the Weierstrass M-test are satisfied.) –  Jonas Meyer Jan 26 '10 at 5:30

1 Answer 1

up vote 8 down vote accepted

An example, due to Fejér, appears in Hille's Analytic function theory on page 122 of the second edition, volume 1.

Erdős wrote a paper showing the existence of classes of examples with many zero coefficients, and in that paper he states that Hardy was the first to find an example of what you're looking for. However, Erdős refers to Landau's book for evidence, and I don't read German, so someone more capable and willing may check that out.

Edit: I initially misnamed the author and book containing the example, as pointed out by Harald Hanche-Olsen.

Update: Coincidentally, while skimming through Sasane's Algebras of holomorphic functions and control theory, I came upon another reference to Hardy's example. This time the reference was to Dienes's The Taylor series. I checked, and sure enough there is Hardy's example in the chapter "The Taylor series on its circle of convergence." I scanned it, along with a page containing a result used in the example. Here it is. (The example starts two thirds of the way down page 464, and the cited "105.IV" is the "IV" that appears on page 441.)

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Actually, the link appears to point to Hille's Analytic function theory, not Titchmarsh. –  Harald Hanche-Olsen Jan 26 '10 at 2:58
    
Oops! I had clicked on links for both books at the same time, and obviously got myself confused (in somewhat of a rush). Thanks, I fixed it. –  Jonas Meyer Jan 26 '10 at 4:56
    
The original paper of Hardy: G. H. Hardy, A theorem concerning Taylor's series, Quart. J. Pure Appl. Math. 44 (1913), 147-160. Another paper, cowritten by Erdős, that gives further info: projecteuclid.org/…. –  Jonas Meyer Mar 14 '10 at 7:33

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