On the multiplicities of the zeros of Laguerre-Pólya functions
HTML articles powered by AMS MathViewer
- by Joe Kamimoto, Haseo Ki and Young-One Kim PDF
- Proc. Amer. Math. Soc. 128 (2000), 189-194 Request permission
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 \[ p(x_{1},\dots ,x_{n})\exp \left (-\sum _{j=1}^{n}x_{j}^{2m_{j}}\right )\] 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
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Thomas Craven, George Csordas, and Wayne Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), no. 2, 405–431. MR 881274, DOI 10.2307/1971315
- J. Kamimoto, On an integral of Hardy and Littlewood, Kyushu J. of Math. 52 (1998), 249–263.
- H. Ki and Y. O. Kim, Proof of the Fourier–Pólya conjecture, preprint.
- Young-One Kim, A proof of the Pólya-Wiman conjecture, Proc. Amer. Math. Soc. 109 (1990), no. 4, 1045–1052. MR 1013971, DOI 10.1090/S0002-9939-1990-1013971-3
- Young-One Kim, Critical points of real entire functions and a conjecture of Pólya, Proc. Amer. Math. Soc. 124 (1996), no. 3, 819–830. MR 1301508, DOI 10.1090/S0002-9939-96-03083-3
- Young-One Kim, Critical points of real entire functions whose zeros are distributed in an infinite strip, J. Math. Anal. Appl. 204 (1996), no. 2, 472–481. MR 1421460, DOI 10.1006/jmaa.1996.0449
- E. Laguerre, Oeuvres I, Gauthier–Villars, Paris, 1898.
- B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
- G. Pólya, Über Annäherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279–295.
- —, On the zeros of an integral function represented by Fourier’s integral, Messenger of Math. 52 (1923), 185–88.
- —, Some problems connected with Fourier’s work on transcendental equations, Quart. J. Math. Oxford Ser. 1 (1930), 21–34.
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1–100.
Additional Information
- Joe Kamimoto
- Affiliation: Department of Mathematics, Kumamoto University, Kumamoto 860, Japan
- MR Author ID: 610515
- 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
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 189-194
- MSC (1991): Primary 30D15, 30D35, 41A30, 43A20
- DOI: https://doi.org/10.1090/S0002-9939-99-04970-9
- MathSciNet review: 1616650