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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pólya class ： ウィキペディア英語版 Pólya class The Pólya class is a set of entire functions satisfying the requirement that if ''E(z)'' is in the class, then:〔("Polya class theory for Hermite-Biehler functions of finite order" ) by Michael Kaltenbäck and Harald Woracek, ''J. London Math. Soc.'' (2) 68.2 (2003), pp. 338–354. DOI: 10.1112/S0024610703004502.〕 *''E(z)'' has no zero (root) in the upper half-plane. *$|E(x+iy)|\backslash ge|E(x-iy)|$ for ''x'' and ''y'' real and ''y'' positive. *$|E(x+iy)|$ is a non-decreasing function of ''y'' for positive ''y''. Every entire function of Pólya class can be expressed as the limit of a series of polynomials having no zeros in the upper half-plane. The product of two functions of Pólya class is also of Pólya class, so the class constitutes a monoid under the operation of multiplication of functions. The Pólya class arises from investigations by Georg Pólya in 1913.〔G. Polya: "Über Annäherung durch Polynome mit lauter reellen Wurzeln", Rend. Circ. Mat. Palermo 36 (1913), 279-295.〕 A de Branges space is defined on the basis of some "weight function" of Pólya class, but with the additional stipulation that the inequality be strict – that is, $|E(x+iy)|>|E(x-iy)|$ for positive ''y''. The Pólya class is a subset of the ''Hermite–Biehler class'', which does not include the third of the above three requirements.〔 A function with no roots in the upper half plane is of Pólya class if and only if two conditions are met: that the nonzero roots ''zn'' satisfy : (with roots counted according to their multiplicity), and that the function can be expressed in the form of a Hadamard product :$z^m\; e^\backslash prod\_n\; \backslash left(1-z/z\_n\backslash right)\backslash exp(z\backslash operatorname\backslash frac)$ with ''c'' real and non-positive and Im ''b'' non-positive. (The non-negative integer ''m'' will be positive if ''E''(0)=0. Even if the number of roots is infinite, the infinite product is well defined and converges.〔Section 7 of the book by de Branges.〕) Louis de Branges showed a connexion between functions of Pólya class and analytic functions whose imaginary part is non-negative in the upper half-plane (UHP), often called Nevanlinna functions. If a function ''E''(''z'') is of Hermite-Biehler class and ''E''(0) = 1, we can take the logarithm of ''E'' in such a way that it is analytic in the UHP and such that log(''E''(0)) = 0. Then ''E''(''z'') is of Pólya class if and only if :$\backslash text\backslash fracz\backslash ge\; 0$ (in the UHP).〔Section 14 of the book by de Branges, or 〕 ==Laguerre–Pólya class== A smaller class of entire functions is the Laguerre–Pólya class, which consists of those functions which are locally the limit of a series of polynomials whose roots are all real. Any function of Laguerre–Pólya class is also of Pólya class. Some examples are $\backslash sin(z),\; \backslash cos(z),\; \backslash exp(z),\; \backslash text\backslash exp(-z^2).$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pólya class」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.