The problem:
Let $g$ be a continuous and $2\pi$ periodic, such that there exists a continuous and $2\pi$ periodic function $f$ which sustains $f*g=g$. Prove that $g$ is a trigonometric polynomial.
Ok. So I know that I need to show somehow that $g(x)=\sum_{n=-\infty}^{\infty}a_ne^{inx}$.
I tried this approach but got stuck:
According to the convolution theorem, we get that the fourier coefs are equal: $$\widehat{(f*g)}(n)=\hat{f}(n)\cdot\hat{g}(n)$$ and from that we get $$\hat{f}(n)\cdot\hat{g}(n)=\hat{g}(n)$$
and that's pretty much where I got stuck.
Any help would be appreciated.
That's it, is it not? $\left(\widehat f(n)\right)_{n\in\Bbb Z}$ is a square-summable sequence by Bessel, and therefore it can't be frequently equal to $1$ as $\lvert n\rvert\to\infty$. Therefore, $\left(\widehat g(n)\right)_{n\in\Bbb Z}$ must be eventually $0$ as $\lvert n\rvert\to\infty$.