Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 2 (2002), No. 2, 449--467 Copyright Heldermann Verlag 2002 A Generalization of Newman's Result on the Zeros of Fourier Transforms Haseo Ki haseo@yonsei.ac.kr, Department of Mathematics, Yonsei University, Seoul 120--749, Korea. Young-One Kim kimyo@math.snu.ac.kr, School of Mathematical Sciences, Seoul National University, Seoul 151--742, Korea. [Abstract-pdf] [Abstract-ps] In this paper, the class of complex Borel measures $\mu$, satisfying $\mu(-E)=\overline{\mu(E)}$ for every Borel set $E\subset\mathbb{R}$, such that the functions $f_{\mu, \lambda}$, $\lambda>0$, defined by $$f_{\mu,\lambda}(z)= \int_{-\infty}^{\infty} \exp\!\left(-\tfrac{\lambda}{2} t^2 + izt\right) \,d\mu(t),$$ have only real zeros, is completely determined. It is done by establishing a general theorem (Theorem \ref{thm:1.3}) on the asymptotic behavior of the zero-distribution of $f_{\mu,\lambda}$ for $\lambda\to\infty$. The theorem is applied to the Riemann $\xi$-function. Keywords: Zeros of Fourier transforms, Riemann's xi-function. MSC 2000: 30C15, 30D10. [FullText-pdf (388 K)] [FullText-ps (352 K)]