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,⋯, are real and simple. Then, using this result, we show that there are infinitely many polynomials p(x₁,⋯, x_n) such that for each (m₁,⋯, m_n) ∈ (ℕ \ {0})ⁿ the translates of the function p(x₁,⋯, x_n)exp (-∑_j=1ⁿx_j^2mj) generate L¹(ℝⁿ). Finally, we discuss the problem of finding the minimum number of monomials pα(x₁,⋯, x_n), α ∈ A, which have the property that the translates of the functions pα(x₁,⋯, x_n)exp(-∑_j=1ⁿx_j^2mj), α ∈ A, generate L¹ℝⁿ), for a given (m₁,⋯,m_n) ∈ (ℕ\{0})ⁿ.