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| 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'''.
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| The concept is named after [[Louis de Branges]] who proved numerous results regarding these spaces, especially as [[Hilbert space]]s, and used those results to prove the [[Bieberbach conjecture]].
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| ==De Branges functions==
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| A '''de Branges function''' is an [[entire function]] ''E'' from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> that satisfies the inequality <math>|E(z)| > |E(\bar z)|</math>, for all ''z'' in the [[upper half-plane|upper half of the complex plane]] <math>\mathbb{C}^+ = \{z \in \mathbb{C} | {\rm Im}(z) > 0\}</math>.
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| ==Definition 1==
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| Given a de Branges function ''E'', the de Branges space ''B''(''E'') is defined as the set of all [[entire function]]s ''F'' such that
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| : <math>F/E,F^{\#}/E \in H_2(\mathbb{C}^+)</math>
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| where:
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| * <math>\mathbb{C}^+ = \{z \in \mathbb{C} | {\rm Im(z)} > 0\}</math> is the open upper half of the complex plane.
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| * <math>F^{\#}(z) = \overline{F(\bar z)}</math>.
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| * <math>H_2(\mathbb{C}^+)</math> is the usual [[Hardy space]] on the open upper half plane.
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| ==Definition 2== | |
| A de Branges space can also be defined as all entire functions ''F'' satisfying all of the following conditions:
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| * <math>\int_{\mathbb{R}} |(F/E)(\lambda)|^2 d\lambda < \infty </math>
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| * <math>|(F/E)(z)|,|(F^{\#}/E)(z)| \leq C_F(\operatorname{Im}(z))^{(-1/2)}, \forall z \in \mathbb{C}^+</math>
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| ==As Hilbert spaces==
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| Given a de Branges space ''B''(''E''). Define the scalar product:
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| :<math>[F,G]=\frac{1}{\pi} \int_{\mathbb{R}} \overline{F(\lambda)} G(\lambda) \frac{d\lambda}{|E(\lambda)|^2}.</math>
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| A de Branges space with such a scalar product can be proven to be a [[Hilbert space]].
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| ==References==
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| * {{cite journal|author=Christian Remling|title=Inverse spectral theory for one-dimensional Schrödinger operators: the A function|journal=Math. Z.|volume=245|year=2003}}
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| [[Category:Operator theory]]
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| [[Category:Hardy spaces]]
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