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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5$\begingroup$ Actually it's true for any continuous functions, it's a classic result I think due to Carleman. mathoverflow.net/questions/26243/… $\endgroup$– Pietro MajerCommented Mar 16, 2015 at 18:16
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2 Answers
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Convolve it with narrower and narrower Gauss kernels.
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1$\begingroup$ Convolutions are smooth, but why analytic? $\endgroup$ Commented Mar 16, 2015 at 18:00
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1$\begingroup$ For bounded function it is well defined and real analytic indeed. $\endgroup$ Commented Mar 16, 2015 at 18:16
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10$\begingroup$ I'm secretly solving the heat equation with the given function as the initial data. $\endgroup$ Commented Mar 16, 2015 at 18:54
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4$\begingroup$ Don't you need uniformly continuous functions to get uniform convergence? $\endgroup$ Commented Mar 17, 2015 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.
answered Mar 17, 2015 at 18:14