On the multiplicities of the zeros of LaguerrePólya functions
Authors:
Joe Kamimoto, Haseo Ki and YoungOne Kim
Journal:
Proc. Amer. Math. Soc. 128 (2000), 189194
MSC (1991):
Primary 30D15, 30D35, 41A30, 43A20
Published electronically:
June 21, 1999
MathSciNet review:
1616650
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We show that all the zeros of the Fourier transforms of the functions , , are real and simple. Then, using this result, we show that there are infinitely many polynomials such that for each the translates of the function generate . Finally, we discuss the problem of finding the minimum number of monomials , , which have the property that the translates of the functions , , generate , for a given .
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 T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the PólyaWiman conjecture, Ann. of Math. (2) 125 (1987), 405431. MR 88a:30007
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 J. Kamimoto, On an integral of Hardy and Littlewood, Kyushu J. of Math. 52 (1998), 249263. CMP 98:09
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 , Critical points of real entire functions whose zeros are distributed in an infinite strip, J. Math. Anal. Appl. 204 (1996), 472481. MR 98e:30030
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 [P2]
 , On the zeros of an integral function represented by Fourier's integral, Messenger of Math. 52 (1923), 18588.
 [P3]
 , Some problems connected with Fourier's work on transcendental equations, Quart. J. Math. Oxford Ser. 1 (1930), 2134.
 [R]
 W. Rudin, Fourier Analysis on Groups, Interscience Publishers, 1962. MR 27:2808
 [W]
 N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1100.
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Additional Information
Joe Kamimoto
Affiliation:
Department of Mathematics, Kumamoto University, Kumamoto 860, Japan
Email:
joe@sci.kumamotou.ac.jp
Haseo Ki
Affiliation:
Department of Mathematics, Yonsei University, Seoul 120749, Korea
Email:
haseo@bubble.yonsei.ac.kr
YoungOne Kim
Affiliation:
Department of Mathematics, Sejong University, Seoul 143–747, Korea
Email:
kimyo@kunja.sejong.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002993999049709
PII:
S 00029939(99)049709
Keywords:
Fourier transform,
LaguerreP\'{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 GrantinAid 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
