Convergence of the series with compactly supported Fourier transform

Question

Let $h $ be an even, real-valued test function $\in C^{\infty}(\mathbb{R})$ in the Schwartz class such that $\widehat h$ has compact support in $[-1,1]$. What can we say about the convergence of the series $ \sum_{n \in \mathbb{Z}} h(M(t+n))$ from the conditions on $h$, where $M$ is a positive integer and $t\in [0,1)$? Is there a value of $t$ for which the series is divergent? If we assume $h$ to be in $L^{1}(\mathbb{R})$, then the result is true by Poisson summation formula. I can't show this from the given conditions on $h$.