reference request - Approximation by analytic functions

Dear all.

Let
$$
f(x) = sum_{k in mathbb{Z}} hat{f}(k) exp(2pi mathrm{i} kx)
$$
be a function given by usual fourier series.

Since my original question hasn't got any answer yet, and I came across another related question, I am just adding it. Denote by $T_n$ the set of all trigonometric polynomials of degree $n$, that is $gin T_n$ if
$$
g(x) = sum_{k=-n}^{n} hat{g}(k) exp(2pi mathrm{i} kx).
$$
So now what is $min_{g in T_n} |f - g|_{infty}$ and what is the optimal $g$?

Since the Fourier series of a continuous function must not converge, I expect that the answer isn't $g(x) = sum_{k=-n}^{n} hat{f}(k) exp(2pi mathrm{i} kx)$ but something else. However, the other choice the Fejer kernel
$$
g(x) = sum_{k=-n}^{n} frac{n - |k|}{n} hat{f}(k) exp(2pi mathrm{i} kx)
$$
seems to give worse estimates on $min_{g in T_n} |f - g|_{infty}$ once $hat{f} in ell^2$.

Thanks,
Helge

Original question:

I am interested in the question of how well one can approximate $f$ by functions that are analytic in some strip. The naive approach yields for example that if one sets
$$
f_R(x) = sum_{|k|leq R} hat{f}(k) exp(2pi mathrm{i} kx)
$$
and assumes $f in C^{n+1}$ then $f_R(x)$ has an extension to a strip of width $frac{n log(k)}{2pi k}$ on which $f_R$ is bounded by $|hat{f}|_{ell^1}$.

This seems like a pretty natural question so I expect it to be well studied, but I don't know where... Does anybody has references? I am also interested in stronger regularity assumptions than $C^n$...