Fourier series via complex analysis

[Mystic998]

Homework Statement

Show that f is 2-pi periodic and analytic on the strip $\vert Im(z) \vert < \eta$, iff it has a Fourier expansion $f(z) = \sum_{n = -\infty}^{\infty} a_{n}z^{n}$, and that $a_n = \frac{1}{2 \pi i} \int_{0}^{2\pi} e^{-inx}f(x) dx$. Also, there's something about the lim sup of the modulus nth roots of the coefficients being less than $e^{-\eta}$. But I think that follows pretty obviously from the Laurent series existing.

Homework Equations

The Attempt at a Solution

The direction assuming it has a Fourier expansion is pretty easy. Just set $\omega = e^{iz}$. Then the series with $\omega$ is analytic on the strip, and that it's 2-pi periodic is clear.

For the other direction I think the idea here is to find a Laurent series for f(z) (or at least that agrees with f on a "big enough" subset of its domain) that's actually analytic on the annulus $e^{-\eta} < \vert z \vert < e^{\eta}$. Then you just use that Laurent series with the variable replaced by $e^{iz}$.

However there doesn't seem to be any way to get such a series that I can see. I've tried a lot of ideas: I've tried looking for a Laurent series for f in the intersection of the annulus with the strip, I've tried finding an appropriate Laurent series for f for an annulus contained in the strip, and I can't get those to work. Also I've tried cheating and just appealing to the Fourier series existing for a real valued function and then using the fact that if analytic functions agree on a "big enough" part of their domain, they have to be the same, but that just seems like a cop out and not very convincing.

So, I'd appreciate any hints or help. Thanks.

 

[mighty2000]

Hello, thank you for posting this interesting problem. After reading your attempt at a solution, here are a few suggestions that may help you make progress:

1. Try using Cauchy's Integral Formula to find a Laurent series for f(z) on the annulus e^{-\eta} < \vert z \vert < e^{\eta}. The key here is to use the fact that f is analytic on the strip \vert Im(z) \vert < \eta to simplify the integral.

2. Consider using the Cauchy-Riemann equations to show that the Laurent series you found in step 1 is actually convergent on the whole annulus e^{-\eta} < \vert z \vert < e^{\eta}. This will show that the series is actually an analytic function on the annulus.

3. Once you have an analytic function on the annulus, try using the fact that it is 2-pi periodic to show that it has a Fourier series expansion. This can be done by writing the function as a sum of its Fourier coefficients and using the fact that they are given by the integral you mentioned in the problem statement.

I hope these suggestions help you make progress on this problem. Good luck!