Paley-Wiener spaces with vanishing conditions and Painlev\'e VI transcendents

Abstract

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.