In
mathematics, a de Branges space (sometimes written De Branges space) is a concept in
functional analysis and is constructed from a de Branges function.
Given a Hermite-Biehler function E, the de Branges space B(E) is defined as the set of all
entire functionsF such that
where:
is the open upper half of the complex plane.
.
is the usual
Hardy space on the open upper half plane.
Definition 2
A de Branges space can also be defined as all entire functions F satisfying all of the following conditions:
Definition 3
There exists also an axiomatic description, useful in operator theory.
As Hilbert spaces
Given a de Branges space B(E). Define the scalar product:
A de Branges space with such a scalar product can be proven to be a
Hilbert space.
References
Christian Remling (2003). "Inverse spectral theory for one-dimensional Schrödinger operators: the A function". Math. Z. 245: 597–617.
doi:
10.1007/s00209-003-0559-2.