  We integrate in the framework of the semilocal trace formula two recent
discoveries on the spectral realization of the zeros of the Riemann zeta
function by introducing a semilocal analogue of the prolate wave operator. The
latter plays a key role both in the spectral realization of the low lying zeros
of zeta--using the positive part of its spectrum--and of their ultraviolet
behavior--using the Sonin space which corresponds to the negative part of the
spectrum. In the archimedean case the prolate operator is the sum of the square
of the scaling operator with the grading of orthogonal polynomials, and we show
that this formulation extends to the semilocal case. We prove the stability of
the semilocal Sonin space under the increase of the finite set of places which
govern the semilocal framework and describe their relation with Hilbert spaces
of entire functions. Finally, we relate the prolate operator to the metaplectic
representation of the double cover of $\operatorname{SL}(2,\mathbb R)$ with the
goal of obtaining (in a forthcoming paper) a second candidate for the semilocal
prolate operator.
