This work is devoted to the comparison of de Branges--Rovnyak $H(b)$ spaces harmonically weighted Dirichlet spaces $\mathcal{ D}_\mu$. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces $\mathcal{ D}_\mu$, when $\mu$ is a finite sum of atoms. This is a generalization of a previous result by Costara--Ransford \cite{ costara2013}: we make no assumptions on the Pythagorean pair $(b,a)$, and we produce new examples.
Categories: math.FA math.CV