Equivalent characterization of zero-sum test function

Question

I am trying to prove an offhand claim from Meyer's Wavelets & Operators, and am stuck on the following. Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported real-valued test functions on the real line. Show that $g\in\mathcal{D}(\mathbb{R})$ satisfies $$\sum_{k=-\infty}^{\infty}g(x+2k\pi)=0$$ if and only if there exists some $h\in\mathcal{D}(\mathbb{R})$ for which $g(x)=h(x+2\pi)-h(x)$. The reverse 'direction' is clear, so it is the forward claim that I'm stuck on. I have a vague feeling that this would follow from a function decomposition-type result from analysis, but I can't quite identify which one to use. I would greatly appreciate any hints (preferably no spoilers!) you guys have to offer.

Answer 1

I don't think this needs a decomposition result but rather an ad hoc approach. The $h$ must satisfy $$ h(x)=-g(x)+h(x+2\pi)=-g(x)-g(x+2\pi)+h(x+4\pi) $$ etc. So by iteration there is only one possible candidate $$ h(x)=-\sum_{k=0}^{\infty} g(x+2k\pi)\ . $$ It is easy to see this is well defined and $C^{\infty}$. What is nontrivial is the finite support property. Note that the hypothesis tells you that $\lim h(x)=0$ at $-\infty$ and $\infty$. What you need is a sharper form of this observation so $h$ has to actually be equal (rather than approximately equal) to zero near $-\infty$ and $\infty$.