Chui's conjecture in Bergman spaces

The talk will be devoted to an analogue of the Chui's problem on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, it will be shown that for every N, the simplest fractions with $N$ poles on the unit circle have minimal norm if and only if the poles are equispaced on the circle. We provide sharp asymptotics of these norms. Finally, it will be demonstrated how the closure of the simplest fractions in weighted Bergman spaces looks like. The talk is based on the joint work by E. Abakumov, A. Borichev, and K. Fedorovskiy (see https://arxiv.org/abs/2009.01898).