de Branges space

Summary

A de Branges space is a Hilbert space of entire functions built from one entire function $E$. The function $E$ must be larger at each point of the upper half-plane than at the mirror point in the lower half-plane. Its elements are the entire functions $F$ such that $F/E$ and $F^\#/E$ lie in the Hardy space of the upper half-plane. The norm of $F$ is the $L^2$ norm of $F/E$ on the real line. It is a reproducing kernel Hilbert space with an explicit kernel, and it generalises the Paley–Wiener spaces. De Branges used these spaces in the spectral theory of canonical systems.