Paley–Wiener spaces with vanishing conditions and Painlevé VI transcendents

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• 作者：Jean-Fran&#xe7 ; ois Burnol^a ; ^{burnol@math.univ-lille1.fr}
• 关键词：Paley– ; Wiener ; Painlev&#xe9 ; Krein
• 刊名：Journal of Functional Analysis
• 出版年：2011
• 出版时间：1 June 2011
• 年：2011
• 卷：260
• 期：11
• 页码：3222-3251
• 全文大小：254 K

文摘

We modify the classical Paley–Wiener spaces PW_x of entire functions of finite exponential type at most x>0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z₁,…,z_n. We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z₁,…,z_n are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions.