For a positive finite Borel measure μ compactly supported in the complex plane, the space \(\mathscr{P}^{2}(\mu)\) is the closure of the analytic polynomials in the Lebesgue space L2(μ). According to Thomson’s famous result, any space \(\mathscr{P}^{2}(\mu)\) decomposes as an orthogonal sum of pieces which are essentially analytic, and a residual L2-space. We study the structure of this decomposition for a class of Borel measures μ supported on the closed unit disk \(\overline{\mathbb{D}}\) for which the part μⅅ, living in the open disk 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 Szegö’s classical theorem.
