# Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$ does $$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$ hold for $$k(p,q)=\sum_{r=0}^{\infty}\sum_{l=r,r-2,\dots,\ge 0}\sum_{m\in\{l,-l\}}c_{r}^{m}\exp\left(-imp\right)\exp\left(imq\right).$$

Fubini seems to impose a far too strong condition as a Fourier series "seldom" converges absolutely. Is it enough that $\sum_{r=0}^{\infty}c_{r}^{m}$ is in $l^2$ in $m$?

• It would be helpful, I think, at least to me, if you filled in some of the details. E.g., is everything happening with periodic functions? Integral on a circle? In what sense "equality"? As distribution? Pointwise? Etc... With mild hypotheses, Fourier series in two variables, on circle $\times$ circle work very well, in the context of Sobolev spaces, which amounts to polynomial-growth constraints on "coefficients"... assuming one can tolerate Fourier expansions which are "merely" distributions, and so on. Clarify, please? – paul garrett Aug 3 '15 at 21:16
• Thank you. The equality should be as $L^2$ functions but other results would be of interest, too. With an appropriate polynomial growth assumption Fubini would be applicable which is what I wanted to avoid. – warsaga Aug 5 '15 at 6:51