Revisiting mean-square approximation by polynomials in the unit disk

For a positive finite Borel measure $\mu$ compactly supported in the complex plane, the space $\mathcal{P}^2(\mu)$ is the closure of the analytic polynomials in the Lebesgue space $L^2(\mu)$. According to Thomson's famous result, any space $\mathcal{P}^2(\mu)$ decomposes as an orthogonal sum of pieces which are essentially analytic, and a residual $L^2$-space. We study the structure of this decomposition for a class of Borel measures $\mu$ supported on the closed unit disk for which the part $\mu_\mathbb{D}$, living in the open disk $\mathbb{D}$, is radial and decreases at least exponentially fast near the boundary of the disk. For the considered class of measures, we give a precise form of the Thomson decompsition. In particular, we confirm a conjecture of Kriete and MacCluer from 1990, which gives an analog to Szeg\"o's classical theorem.