Computational Methods and Function Theory 2 (2002), No. 2, 449--467
Copyright Heldermann Verlag 2002
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.
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.
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