# Periodic solutions of $f^{(m)}=g$

Let $g:R \rightarrow R$ be continuous and $2\pi$-periodic, let $m \in N$. How many solution in class of $m$-times continuously differentiable $2\pi$-periodic functions has equation $$f^{(m)}=g ?$$

Edit. Obviously, if $f$ is a solution in this class and $C$ is a constant then $f+C$ is also a solution. Are there another solutions?

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More generally if $f$ is a solution and $p$ a polynomial of degree less than $m$ then $f + p$ is a solution, and conversely if $f_1, f_2$ are two solutions then $f_1 - f_2$ is a polynomial of degree less than $m$. So once you find a single solution (e.g. by the method below) you have all solutions. –  Qiaochu Yuan May 3 '12 at 15:55

If $g$ is $2\pi$-periodic then you can use Fourier series and write $$g(x)=\sum_{k=-\infty}^\infty g_ke^{i k x}$$ and so also $$f(x) = \sum_{k=-\infty}^\infty f_ke^{i k x}.$$ By direct substitution you get $$\sum_{k=-\infty}^\infty (ik)^m f_k e^{i k x}=\sum_{k=-\infty}^\infty g_k e^{i k x}$$ then, all the solutions that can be cast in the form $$(ik)^mf_k=g_k$$ are periodic solutions of the given equation.