6
$\begingroup$

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 $C^n$...

$\endgroup$
7
  • 2
    $\begingroup$ 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) $\endgroup$ Jun 20, 2010 at 12:19
  • $\begingroup$ 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. $\endgroup$
    – Helge
    Jun 20, 2010 at 13:58
  • $\begingroup$ 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).... $\endgroup$
    – Zen Harper
    Jun 21, 2010 at 4:34
  • $\begingroup$ ....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. $\endgroup$
    – Zen Harper
    Jun 21, 2010 at 4:38
  • $\begingroup$ 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 $sup_{x \in [0,1], 0 < y < \rho} |f(x+iy)| < 100$. $\endgroup$
    – Helge
    Jun 21, 2010 at 11:50

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ I think this should be helpful. At least, I have some idea what to look at now :-). $\endgroup$
    – Helge
    Jun 21, 2010 at 20:49
  • $\begingroup$ 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. $\endgroup$ Jun 21, 2010 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.