Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $(m_{1},\dots , m_{n})\in (\mathbb{N}\setminus \{0\})^{n}$.

References [Enhancements On Off] (What's this?)

  • [B] N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197-226. MR 12:250a
  • [CCS] T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405-431. MR 88a:30007
  • [K] J. Kamimoto, On an integral of Hardy and Littlewood, Kyushu J. of Math. 52 (1998), 249-263. CMP 98:09
  • [KK] H. Ki and Y. O. Kim, Proof of the Fourier-Pólya conjecture, preprint.
  • [Km1] Y. O. Kim, A proof of the Pólya-Wiman conjecture, Proc. Amer. Math. Soc. 109 (1990), 1045-1052. MR 90k:30049
  • [Km2] -, Critical points of real entire functions and a conjecture of Pólya, Proc. Amer. Math. Soc. 124 (1996), 819-830. MR 96f:30027
  • [Km3] -, Critical points of real entire functions whose zeros are distributed in an infinite strip, J. Math. Anal. Appl. 204 (1996), 472-481. MR 98e:30030
  • [L] E. Laguerre, Oeuvres I, Gauthier-Villars, Paris, 1898.
  • [Le] B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Mono., vol. 5, A.M.S., Providence, R.I., 1964. MR 81k:30011
  • [P1] G. Pólya, Über Annäherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279-295.
  • [P2] -, On the zeros of an integral function represented by Fourier's integral, Messenger of Math. 52 (1923), 185-88.
  • [P3] -, Some problems connected with Fourier's work on transcendental equations, Quart. J. Math. Oxford Ser. 1 (1930), 21-34.
  • [R] W. Rudin, Fourier Analysis on Groups, Interscience Publishers, 1962. MR 27:2808
  • [W] N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1-100.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D15, 30D35, 41A30, 43A20

Retrieve articles in all journals with MSC (1991): 30D15, 30D35, 41A30, 43A20

Additional Information

Joe Kamimoto
Affiliation: Department of Mathematics, Kumamoto University, Kumamoto 860, Japan

Haseo Ki
Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Korea

Young-One Kim
Affiliation: Department of Mathematics, Sejong University, Seoul 143–747, Korea

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

American Mathematical Society