---
tag: Y7JP4NA725
kind: definition
name: de Branges space
msc: ["46E22", 30D20]
defines: NYHNTP9AND
uses: [0419794R0T, X6XGGH5FPW, PNNNFJVF6W]
sources:
  - url: https://en.wikipedia.org/wiki/De_Branges_space
---

## Definition

Let $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ be the [[upper half-plane|X6XGGH5FPW]].
For an [[entire function|0419794R0T]] $F$ let $F^\#(z) = \overline{F(\bar z)}$, which is again entire.
Let $E$ be an entire function such that $\abs{E(\bar z)} < \abs{E(z)}$ for all $z \in \mathbb{H}$.
The *de Branges space* $\mathcal{H}(E)$ is the set of entire functions $F$ such that

1. $F/E$ belongs to the [[Hardy space|PNNNFJVF6W]] $H^2(\mathbb{H})$, and
2. $F^\#/E$ belongs to $H^2(\mathbb{H})$,

with the inner product
$$\ip{F}{G} = \int_{-\infty}^{\infty} \frac{F(x) \overline{G(x)}}{\abs{E(x)}^2} \, dx.$$

## Remarks

$\mathcal{H}(E)$ is a [[Hilbert space|HM2HHE3J51]] and a [[reproducing kernel Hilbert space|4CK2GSRAGD]].
For $w \in \C$ the reproducing kernel is
$$K_w(z) = \frac{\overline{E(w)} E(z) - \overline{E^\#(w)} E^\#(z)}{2\pi i (\bar w - z)},$$
so that $F(w) = \ip{F}{K_w}$ for all $F \in \mathcal{H}(E)$.
The condition on $E$ implies that $E$ has no zeros in $\mathbb{H}$, so condition 1 makes sense.

Example: $E(z) = e^{-iaz}$ with $a > 0$.
For $z = x + iy$ with $y > 0$ one has $\abs{E(z)} = e^{ay} > e^{-ay} = \abs{E(\bar z)}$.
Here $E^\#(z) = e^{iaz}$, and the kernel becomes $K_w(z) = \frac{\sin(a(\bar w - z))}{\pi (\bar w - z)}$.
$\mathcal{H}(E)$ is the Paley–Wiener space of entire functions of exponential type at most $a$ that are square integrable on $\R$.

Non-example: $E(z) = 1$ gives $\abs{E(\bar z)} = \abs{E(z)}$, so the strict inequality fails and $E$ does not define a de Branges space.