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  "body": "\n## Definition\n\nLet $\\mathbb{H} = \\set{z \\in \\C : \\operatorname{Im} z \u003e 0}$ be the [[upper half-plane|X6XGGH5FPW]].\nFor an [[entire function|0419794R0T]] $F$ let $F^\\#(z) = \\overline{F(\\bar z)}$, which is again entire.\nLet $E$ be an entire function such that $\\abs{E(\\bar z)} \u003c \\abs{E(z)}$ for all $z \\in \\mathbb{H}$.\nThe *de Branges space* $\\mathcal{H}(E)$ is the set of entire functions $F$ such that\n\n1. $F/E$ belongs to the [[Hardy space|PNNNFJVF6W]] $H^2(\\mathbb{H})$, and\n2. $F^\\#/E$ belongs to $H^2(\\mathbb{H})$,\n\nwith the inner product\n$$\\ip{F}{G} = \\int_{-\\infty}^{\\infty} \\frac{F(x) \\overline{G(x)}}{\\abs{E(x)}^2} \\, dx.$$\n\n## Remarks\n\n$\\mathcal{H}(E)$ is a [[Hilbert space|HM2HHE3J51]] and a [[reproducing kernel Hilbert space|4CK2GSRAGD]].\nFor $w \\in \\C$ the reproducing kernel is\n$$K_w(z) = \\frac{\\overline{E(w)} E(z) - \\overline{E^\\#(w)} E^\\#(z)}{2\\pi i (\\bar w - z)},$$\nso that $F(w) = \\ip{F}{K_w}$ for all $F \\in \\mathcal{H}(E)$.\nThe condition on $E$ implies that $E$ has no zeros in $\\mathbb{H}$, so condition 1 makes sense.\n\nExample: $E(z) = e^{-iaz}$ with $a \u003e 0$.\nFor $z = x + iy$ with $y \u003e 0$ one has $\\abs{E(z)} = e^{ay} \u003e e^{-ay} = \\abs{E(\\bar z)}$.\nHere $E^\\#(z) = e^{iaz}$, and the kernel becomes $K_w(z) = \\frac{\\sin(a(\\bar w - z))}{\\pi (\\bar w - z)}$.\n$\\mathcal{H}(E)$ is the Paley–Wiener space of entire functions of exponential type at most $a$ that are square integrable on $\\R$.\n\nNon-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.\n",
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