De Branges functions
A Hermite-Biehler function, also known as de Branges function is an entire function E from to that satisfies the inequality , for all z in the upper half of the complex plane .
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 to that satisfies the inequality , for all z in the upper half of the complex plane .
Given a Hermite-Biehler function E, the de Branges space B(E) is defined as the set of all entire functions F such that
where:
A de Branges space can also be defined as all entire functions F satisfying all of the following conditions:
There exists also an axiomatic description, useful in operator theory.
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.
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