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A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called $\mathbb{R}$-analytic iff for every $x_0 \in \mathbb{R} $ there exist $R>0$ and power series $\sum_{n=0}^\infty a_n (x-x_0)^n$ convergent for $|x-x_0| and such that $f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$ for $|x-x_0|.

For some strange continuous functions on $\mathbb{R}$, for example for Weierstrass continuous nondifferentiable function (i.e. $f(x)=\sum_{n=0}^\infty a^n \cos(b^n \pi x)$ for $x\in \mathbb{R}$, where $0, $b$ is positive odd integer such that ab> 1+\frac{3}{2}\pi), there exist a sequence of $\mathbb{R}$-analytic functions (even entire functions) which converges to $f$ uniformly (in the case of Weierstrass function it is sufficient to take the sequence of partial sum of series defining this function).

Is it maybe true that for every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ there exists a sequence of $\mathbb{R}$ analytic functions which converges uniformly to $f$ ?

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    @MartinArgerami Yes I misunderstood the question.2012-02-11

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The proof of the assertion seems to be contained in the paper

Carleman, T., Sur un théorème de Weierstrass, Ark. Mat., Ast. Fysik 20B (1927), 1-5

a PDF of which is available (in French) from this page. In fact, he proves a bit more. A plain explanation can be found in other papers (e.g., here; I would post more links if I had more reputation). What I gather that he proved is, given a continuous function $f$ and a continuous notion of acceptable "error" (with error $\epsilon$ coming from choosing the constant function $\epsilon$), one can find an entire function approximating to the given precision.

A nice proof is reported to be contained in Lectures on Complex Approximation Theory by D. Gaier, Birkhaussen 1987 on page 49.

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    Skimming the Carleman paper, I don't immediately see how to get the "better than uniform" approximation. Carleman only seems to discuss uniform approximation. I don't see how Boghossian extracts the claimed statement.2017-12-11