In this paper, we study the function spaces \({\cal D}(\mu)\) by Richter (1991) and Aleman (1993), and \({{\cal D}_{\vec \mu}}\) by Rydhe (2019). It is known that the forward shift Mz is bounded and expansive on \({\cal D}(\mu)\) , and therefore \({\cal D}(\mu)\) coincides with a de Branges-Rovnyak space \({\cal H}[B]\) . We show that such a B is rational if and only if μ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of \({\cal D}(\mu)\) for finitely atomic μ. Similarly, we characterize the allowable tuples \({\vec \mu}=({\vert dz \vert \over 2 \pi}, \mu_{1},\cdots,\mu_{n-1})\) such that Mz on \({\cal D}_{\vec \mu}\) is expansive with a finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples \({\vec \mu}\) .
