# Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents - Mathematics > Classical Analysis and ODEs

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Abstract: We modify the classical Paley-Wiener spaces $PW x$ of entire functions offinite exponential type at most $x>0$, which are square integrable on the realline, via the additional condition of vanishing at finitely many complex points$z 1, ., z n$. We compute the reproducing kernels and relate their variationswith respect to $x$ to a Krein differential system, whose coefficient which wecall the $\mu$-function and solutions have determinantal expressions.Arguments specific to the case where the -trivial zeros- $z 1, ., z n$ are inarithmetic progression on the imaginary axis allow us to establish forexpressions arising in the theory a system of two non-linear first orderdifferential equations. A computation, having this non-linear system at hisstart, obtains quasi-algebraic and among them rational Painlev\-e transcendentsof the sixth kind as certain quotients of such $\mu$-functions.

Autor: Jean-François Burnol

Fuente: https://arxiv.org/