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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=...