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.
The concept is named after Louis de Branges who proved numerous results regarding these spaces, especially as Hilbert spaces, and used those results to prove the Bieberbach conjecture.
A Hermite-Biehler function, also known as de Branges function is an entire function E from
\Complex
\Complex
|E(z)|>|E(\barz)|
\Complex+=\{z\in\Complex\mid\operatorname{Im}(z)>0\}
Given a Hermite-Biehler function, the de Branges space is defined as the set of all entire functions F such thatwhere:
\Complex+=\{z\in\Complex\mid\operatorname{Im}(z)>0\}
F\#(z)=\overline{F(\barz)}
+) | |
H | |
2(\Complex |
A de Branges space can also be defined as all entire functions satisfying all of the following conditions:
\int\Reals|(F/E)(λ)|2dλ<infty
|(F/E)(z)|,|(F\#/E)(z)|\leq
(-1/2) | |
C | |
F(\operatorname{Im}(z)) |
,\forallz\in\Complex+
There exists also an axiomatic description, useful in operator theory.
Given a de Branges space . Define the scalar product:
^2 |
A de Branges space with such a scalar product can be proven to be a Hilbert space.