On the multiplicities of the zeros of Laguerre-Polya 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 t hat there are infinitely many polynomials p(x(1),...,x(n)) such that for ea ch (m(1),...,m(n)) is an element of (N\{0})(n) the translates of the functi on [GRAPHICS] generate L-1 (R-n). Finally, we discuss the problem of finding the minimum number of monomials p(alpha)(x(1),...,x(n)), alpha is an element of A, whic h have the property that the translates of the functions p(alpha)(x(1),..., x(n)) exp(-Sigma(j=1)(n) x(j)(2mj)), alpha is an element of A, generate L-1 (R-n), for a given (m(1),...,m(n)) is an element of (N\{0})(n).