On the multiplicities of the zeros of Laguerre-Pólya functions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the multiplicities of the zeros of Laguerre-Pólya functions

Authors:	Joe Kamimoto Haseo Ki Young-One Kim

Affiliation:	Department of Mathematics, Kumamoto University, Kumamoto 860, Japan ; Department of Mathematics, Yonsei University, Seoul 120-749, Korea ; Department of Mathematics, Sejong University, Seoul 143--747, Korea

Abstract:	We show that all the zeros of the Fourier transforms of the functions $exp (-x^{2m})$ , , 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 $p_{alpha }(x_{1},dots , x_{n})$ , , which have the property that the translates of the functions $p_{alpha }(x_{1},dots , x_{n})exp (-sum _{j=1}^{n}x_{j}^{2m_{j}})$ , , generate $L^{1}(mathbb{R}^{n})$ , for a given $(m_{1},dots , m_{n})in (mathbb{N}setminus {0})^{n}$ .

Keywords:	Fourier transform Laguerre--P'{o}lya function Wiener's theorem

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