Density of simple partial fractions with poles on the circle in weighted spaces for the disk and the interval

Abstract

Approximation properties of simple partial fractions (the logarithmic derivatives of algebraic polynomials) having all poles on the unit circle are investigated. We obtain criteria for the density of such fractions is some classical integral spaces: in the spaces of functions summable with degree p in the unit interval with the ultraspheric weight and in the (weighted) Bergman spaces of functions analytic in the unit disk and summable with degree p over the area of the disk. Our results generalize to the case of an arbitrary exponent p > 0 the known criteria by Chui and Newman and by Abakumov, Borichev and Fedorovskiy for the Bergman spaces with p = 1 and p = 2, correspondingly.