I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ such that $$|\partial^\alpha_x \partial^\beta_\xi a(x, \xi)|\leq C_{\alpha, \beta} \langle \xi\rangle^{M(\alpha, \beta)},$$ for every pair of multi-indices $\alpha, \beta$. How is this condition related to the convergence of the series $$\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi),$$ where $f\in\mathcal{S}(\mathbb R^n)$?
The condition $$|\partial^\alpha_x \partial^\beta_\xi a(x, \xi)|\leq C_{\alpha, \beta} \langle \xi\rangle^{M(\alpha, \beta)},$$ showed up in the hypothesis of a theorem that I should prove and I didn't see how to use this in the proof, so I conjecture it must be for assuring the convergence of the above series..
Any help will be valuable.. Thanks.
Some Definitions:
1. We define $\langle \xi\rangle:=(1+|\xi|^2)^{1/2}$.
2. $\displaystyle f\in\mathcal{S}(\mathbb R^n)\Leftrightarrow \sup_{x\in\mathbb R^n}|x^\beta \partial^\alpha f(x)|<\infty \forall \alpha, \beta\in\mathbb N_0^n.$ The space $\mathcal{S}(\mathbb R^n)$ is called Schwartz space.
** 3.** The convergence of the series $\displaystyle \sum_{\xi\in\mathbb Z^n}e^{2\pi ix\cdot \xi}a(x, \xi) \hat{f}(\xi)$ is in the sense there exists the limit $\displaystyle \lim_{k\to \infty} \sum_{|\xi|\leq k} e^{2\pi ix\cdot \xi} a(x, \xi) \hat{f}(\xi)$.