# Is it true that for each bounded continuous function we can find a set of analytic functions to uniformly converge it?

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 $f$ ?

• Actually it's true for any continuous functions, it's a classic result I think due to Carleman. mathoverflow.net/questions/26243/… Mar 16 '15 at 18:16

Convolve it with narrower and narrower Gauss kernels.

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

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.