Approximation by analytic functions

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 $C^n$...

• I find the question a bit unclear: are you wondering how well the Fourier polynomials approximate $f$ , or are you asking how can one approximate $f$ by analytic functions? (in which case there are other solutions than the Fourier polynomials) – Pietro Majer Jun 20 '10 at 12:19
• I am wondering, what the "best" way to do it. For example one wants to have a not too large extension into a not too small strip. So I would probably say I am interested in other ways than Fourier polynomials. – Helge Jun 20 '10 at 13:58
• This question looks like it could be interesting, but as written it's unclear and vague to me; what kind of strip? Vertical? Horizontal? Do you want uniform approximation, or some other norm? What conditions on the analytic extension do you want? Your comment about f_R(x) seems strange: first, the "width" (or do you mean "height"?) should depend on R, but you've written it with k. Second, f_R is just a finite sum for fixed R, so is entire (and in fact is bounded on each fixed horizontal strip).... – Zen Harper Jun 21 '10 at 4:34
• ....Third, I don't see why you're estimating f_R in terms of the l^1 norm of the Fourier coefficients, when the relevant quantity seems to be the sup norm of the (n+1)th derivative of f. I also don't see how you get that estimate. Do you have a specific problem or function? It might be helpful to give more details. – Zen Harper Jun 21 '10 at 4:38
• Hi Zen. The problem is that I am not sure what the right question is. It is a little puzzle piece in something, I am working on, and I was just hoping somebody would say "There is this great book by ... You will find more than you ever wanted to know there." Maybe a better way to phrase the question is: Given a one bounded function $f$ find a function $g$ such that one minimizes $\|f - g\|_{L^{\infty}([0,1])}$ and maximizes $\rho > 0$ such that $sup_{x \in [0,1], 0 < y < \rho} |f(x+iy)| < 100$. – Helge Jun 21 '10 at 11:50

The answer to the modified question is given by Jackson-type theorems.

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.

• I think this should be helpful. At least, I have some idea what to look at now :-). – Helge Jun 21 '10 at 20:49
• Helge, the book by Akhiezer is certainly very good but it's not the only place to read about optimal harmonic approximations. The keyword is constructive function theory. – Andrey Rekalo Jun 21 '10 at 20:53