On the coefficient formula for de Branges-Rovnyak norms

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^{\mathrm{\infty }}$, and let $\phi =b\mathrm{/}a$, 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 $\phi$ via the formula $$\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathcal{H}(b)}^2=\sum _{m\ge 0}\mathrm{\mid }\widehat{f}(m){\mathrm{\mid }}^2+\sum _{m\ge 0}\mid \sum _{n\ge 0}\overline{\widehat{\phi }(n)}\widehat{f}(m+n){\mid }^2\mathrm{.}$$ 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 $\phi \in H^2$ or $\phi$ is rational. If merely $\phi \in H^p$ for some $p\in (0,2]$, then the formula still holds provided that, in addition, ${\sum }_{m\ge 0}{m}^{2\mathrm{/}p-1}\mathrm{\mid }\widehat{f}(m){\mathrm{\mid }}^2<\mathrm{\infty }$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\in \mathcal{H}(b)$.