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 $C^n$...



Approximation by analytic functions
The answer to the modified question is given by [Jackson-type theorems](http://en.wikipedia.org/wiki/Jackson%27s_inequality).


The classic book by N.I. Akhiezer which is quoted in the Wikipedia article contains a number of specialised results on optimal approximation by trigonometric polynomials. 


A typical optimal result improves the approximation by finite Fourier sums by a logarithmic factor.



> 
> **Theorem.** Let $f$ be a periodic function on $\mathbb R$ of class $C^{m}$. Then for any $n\in\mathbb N$
>  $$\inf\limits\_{g \in T\_n} \|f - g\|\_{L^\infty}\leq \frac{K\_m}{n^m}\|f^{(m)}\|\_{L^\infty},$$
>  where the constant $K\_m$ is sharp (and can be written in a closed form).
> 
> 
> 


A result of S. Bernstein says roughly that the order of approximation $n^{-m}$ cannot be improved.


To find the trigonometric polynomial $g$ which minimizes $\|f - g\|\_{L^\infty}$ for a given $f$ is a difficult problem. I am not sure if it has been solved.


