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Is it true that for each bounded continuous function $f:\mathbb R \to \mathbb R$, we can find a set of analytic functions $g_i:\mathbb R \to \mathbb R, i=1,2,...$ such that $g_i$ uniformly converges to $f$ ?

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Convolve it with narrower and narrower Gauss kernels.

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    $\begingroup$ Convolutions are smooth, but why analytic? $\endgroup$ Mar 16 '15 at 18:00
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    $\begingroup$ Because the Gauss kernel is. $\endgroup$
    – Fan Zheng
    Mar 16 '15 at 18:06
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    $\begingroup$ For bounded function it is well defined and real analytic indeed. $\endgroup$ Mar 16 '15 at 18:16
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    $\begingroup$ I'm secretly solving the heat equation with the given function as the initial data. $\endgroup$
    – Fan Zheng
    Mar 16 '15 at 18:54
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    $\begingroup$ Don't you need uniformly continuous functions to get uniform convergence? $\endgroup$ Mar 17 '15 at 7:43
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In the paper

  • MR0098847 (20 #5299) Grauert, Hans: On Levi's problem and the imbedding of real-analytic manifolds. Ann. of Math. (2) 68 1958 460–472.

it is proved (Proposition 8) that real analytic functions are dense in continuous functions for the Whitney $C^0$-topology, for any paracompact real analytic manifold. The sup-norm gives a coarser topology, so this also follows.

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