Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

Details Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

2010-08-03

arxiv.org/abs/1008.0617v1

Journal of Functional Analysis, Volume 260, Issue 11, 1 June 2011, Pages 3222-3251

Mathematics

Classical Analysis and ODEs

30 pages

Scientific paper

10.1016/j.jfa.2010.12.002

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 $z_1, ..., z_n$. We compute the reproducing kernels and relate their variations with respect to $x$ to a Krein differential system, whose coefficient (which we call the $\mu$-function) and solutions have determinantal expressions. Arguments specific to the case where the "trivial zeros" $z_1, ..., z_n$ 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 Painlev\'e transcendents of the sixth kind as certain quotients of such $\mu$-functions.

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