Dear all.

Let $$ f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx) $$ 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 $g\in T_n$ if $$ g(x) = \sum_{k=-n}^{n} \hat{g}(k) \exp(2\pi \mathrm{i} kx). $$ 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 $g(x) = \sum_{k=-n}^{n} \hat{f}(k) \exp(2\pi \mathrm{i} kx)$ but something else. However, the other choice the Fejer kernel $$ g(x) = \sum_{k=-n}^{n} \frac{n - |k|}{n} \hat{f}(k) \exp(2\pi \mathrm{i} kx) $$ 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 $$ f_R(x) = \sum_{|k|\leq R} \hat{f}(k) \exp(2\pi \mathrm{i} kx) $$ and assumes $f \in C^{n+1}$ then $f_R(x)$ has an extension to a strip of width $\frac{n \log(k)}{2\pi k}$ 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 $C^n$...