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 $m$?