A Generalization of Newman’s Result on the Zeros of Fourier Transforms

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 f _μ,λ, λ > 0, defined by

$$f_{\mu,\lambda}(z)=\int_{-\infty}^{\infty}{\rm exp}(-{\lambda\over 2} t^2 + izt)\ d\mu(t)$$

, 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 f _μλ} for λ → ∞. The theorem is applied to the Riemann ξ-function.