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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the multiplicities of the zeros
of Laguerre-Pólya functions


Authors: Joe Kamimoto, Haseo Ki and Young-One Kim
Journal: Proc. Amer. Math. Soc. 128 (2000), 189-194
MSC (1991): Primary 30D15, 30D35, 41A30, 43A20
Published electronically: June 21, 1999
MathSciNet review: 1616650
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Abstract: We show that all the zeros of the Fourier transforms of the functions $\exp (-x^{2m})$, $m=1,2,\dots $, are real and simple. Then, using this result, we show that there are infinitely many polynomials $p(x_{1},\dots ,x_{n})$ such that for each $(m_{1},\dots , m_{n})\in (\mathbb{N}\setminus \{0\})^{n}$ the translates of the function

\begin{displaymath}p(x_{1},\dots ,x_{n})\exp \left(-\sum _{j=1}^{n}x_{j}^{2m_{j}}\right)\end{displaymath}

generate $L^{1}(\mathbb{R}^{n})$. Finally, we discuss the problem of finding the minimum number of monomials $p_{\alpha }(x_{1},\dots , x_{n})$, $\alpha \in A$, which have the property that the translates of the functions $p_{\alpha }(x_{1},\dots , x_{n})\exp (-\sum _{j=1}^{n}x_{j}^{2m_{j}})$, $\alpha \in A$, generate $L^{1}(\mathbb{R}^{n})$, for a given $(m_{1},\dots , m_{n})\in (\mathbb{N}\setminus \{0\})^{n}$.


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Additional Information

Joe Kamimoto
Affiliation: Department of Mathematics, Kumamoto University, Kumamoto 860, Japan
Email: joe@sci.kumamoto-u.ac.jp

Haseo Ki
Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Korea
Email: haseo@bubble.yonsei.ac.kr

Young-One Kim
Affiliation: Department of Mathematics, Sejong University, Seoul 143–747, Korea
Email: kimyo@kunja.sejong.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04970-9
PII: S 0002-9939(99)04970-9
Keywords: Fourier transform, Laguerre--P\'{o}lya function, Wiener's theorem
Received by editor(s): February 2, 1998
Received by editor(s) in revised form: March 16, 1998
Published electronically: June 21, 1999
Additional Notes: The first author was partially supported by Grant-in-Aid for Scientific Research (No. 10740073), Ministry of Education, Science and Culture, Japan
The second author was supported by Yonsei University Research Fund of 1998
The third author was supported by the Korea Science and Engineering Foundation(KOSEF) through the Global Analysis Research Center(GARC) at Seoul National University.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society