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 $f$ ?

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 Majer Mar 16 '15 at 18:16
Convolve it with narrower and narrower Gauss kernels.

1$\begingroup$ Convolutions are smooth, but why analytic? $\endgroup$ – Fedor Petrov Mar 16 '15 at 18:00

2

1$\begingroup$ For bounded function it is well defined and real analytic indeed. $\endgroup$ – Fedor Petrov Mar 16 '15 at 18:16

9$\begingroup$ I'm secretly solving the heat equation with the given function as the initial data. $\endgroup$ – Fan Zheng Mar 16 '15 at 18:54

4$\begingroup$ Don't you need uniformly continuous functions to get uniform convergence? $\endgroup$ – Jochen Wengenroth Mar 17 '15 at 7:43
In the paper
 MR0098847 (20 #5299) Grauert, Hans: On Levi's problem and the imbedding of realanalytic 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 supnorm gives a coarser topology, so this also follows.