Density of Simple Partial Fractions with Poles on a Circle in Weighted Spaces for a Disk and a Segment

In this paper, we study approximation properties of simple partial fractions (logarithmic derivatives of algebraic polynomials), all of whose poles lie on the unit circle. There are obtained criteria for the density of these fractions in classical integral spaces: in the spaces of functions summable with degree p on the unit segment with ultraspherical weight and (weighted) Bergman spaces, analytic in the unit disk and summable with degree p over the disk area. The well-known criteria of Chui and Newman and Abakumov, Borichev, and Fedorovsky for Bergman spaces with p = 1 and p = 2, respectively, are generalized by the obtained results to the case of an arbitrary exponent p > 0.