On the multiplicities of the zeros of Laguerre-Pólya functions

We show that all the zeros of the Fourier transforms of the functions $\exp (-x^{2m})$, $m=1,2,\dots$, are real and simple. Then, using this result, we show that there are infinitely many polynomials $p(x_{1},\dots ,x_{n})$ such that for each $(m_{1},\dots , m_{n})\in (\mathbb{N}\setminus \{0\})^{n}$ the translates of the function

generate $L^{1}(\mathbb{R}^{n})$. Finally, we discuss the problem of finding the minimum number of monomials $p_{\alpha }(x_{1},\dots , x_{n})$, $\alpha \in A$, which have the property that the translates of the functions $p_{\alpha }(x_{1},\dots , x_{n})\exp (-\sum _{j=1}^{n}x_{j}^{2m_{j}})$, $\alpha \in A$, generate $L^{1}(\mathbb{R}^{n})$, for a given $(m_{1},\dots , m_{n})\in (\mathbb{N}\setminus \{0\})^{n}$.