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It is well known that a sequence (or a series) of holomorphic functions converging uniformly converges to a holomorphic function. I would like to know under what condition a sequence (or a series) or real-analytic functions converges to a real-analytic function. Are there any simple criteria?

In particular, I would like to know whether a series of real-analytic functions converging absolutely and uniformly converges to a real-analytic function. I read in the proof of Lemma 1.8 in "Mock theta functions" written by S. P. Zwegers "since R is the (infinite) sum of real-analytic functions, and the series converges absolutely and uniformly, it is real-analytic." But I cannot understand why that fact holds.

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  • $\begingroup$ Yes, that doesn't make sense. I guess, I was being very sloppy, there... $\endgroup$ – Sander Zwegers Jul 25 '16 at 21:37
  • $\begingroup$ Thank you for your comment. I will check the real-analyticity on a case-by-case basis. $\endgroup$ – user356126 Jul 26 '16 at 13:02
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A uniform limit of real-analytic functions certainly need not be real-analytic. Any continuous function on $[a,b]$ is a uniform limit of polynomials (and is hence the sum of a uniformly and absolutely convergent series of polynomials).

I can't think of any "simple" criteria. Even uniform convergence of a sequence of functions together with uniform convergence of every derivative is not enough; consider the Poisson integral of a function in $C^\infty_c$.

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  • $\begingroup$ Thank you for your comment. The counterexample of the continuous function on [a, b] is simple and interesting. $\endgroup$ – user356126 Jul 26 '16 at 12:59
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This is true in $\mathbb{C}$ but not in $\mathbb{R}$. The reason is that uniform convergence preserves continuity, not differentiability and beyond.

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  • $\begingroup$ Thank you for your answer. I sometimes deal with real-analytic functions. I feel larger difficuty in dealing with real-analytic functions than with holomorphic functions. $\endgroup$ – user356126 Feb 24 '17 at 2:13

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