A generalization of Newman’s result on the zeros of Fourier transforms, Comput. Methods Funct. Theory 2

Abstract. In this paper, the class of complex Borel measures µ, satisfying µ(−E) = µ(E) for every Borel set E ⊂ R, such that the functions fµ,λ, λ> 0, defined by fµ,λ(z) = exp (−λ2 t2 + izt) dµ(t), have only real zeros, is completely determined. It is done by establishing a general theorem (Theorem 1.3) on the asymptotic behavior of the zero-distribution of fµ,λ for λ → ∞. The theorem is applied to the Riemann ξ-function