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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.30114 |
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| _version_ | 1866917544383741952 |
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| author | Ransford, Thomas |
| author_facet | Ransford, Thomas |
| contents | Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $ϕ=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to $\mathcal{H}(b)$, and its norm in $\mathcal{H}(b)$ can be expressed in terms of the Taylor coefficients of $f$ and $ϕ$ via the formula \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2. \] However, the formula can break down for some other $f\in\mathcal{H}(b)$.
In this article we extend the scope of the formula to all $f\in H^2$ for which the right-hand side is finite, provided that either $ϕ\in H^2$ or $ϕ$ is rational. If merely $ϕ\in H^p$ for some $p\in(0,2]$, then the formula still holds provided that, in addition, $\sum_{m\ge0}m^{2/p-1}|\hat{f}(m)|^2<\infty$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\in\mathcal{H}(b)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30114 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the coefficient formula for de Branges-Rovnyak norms Ransford, Thomas Complex Variables Functional Analysis Primary 47B32, Secondary 47B35 Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $ϕ=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to $\mathcal{H}(b)$, and its norm in $\mathcal{H}(b)$ can be expressed in terms of the Taylor coefficients of $f$ and $ϕ$ via the formula \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2. \] However, the formula can break down for some other $f\in\mathcal{H}(b)$. In this article we extend the scope of the formula to all $f\in H^2$ for which the right-hand side is finite, provided that either $ϕ\in H^2$ or $ϕ$ is rational. If merely $ϕ\in H^p$ for some $p\in(0,2]$, then the formula still holds provided that, in addition, $\sum_{m\ge0}m^{2/p-1}|\hat{f}(m)|^2<\infty$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\in\mathcal{H}(b)$. |
| title | On the coefficient formula for de Branges-Rovnyak norms |
| topic | Complex Variables Functional Analysis Primary 47B32, Secondary 47B35 |
| url | https://arxiv.org/abs/2605.30114 |