Approximation by simplest fractions and Chui's conjecture in weighted Bergman spaces

We will discuss the question on approximation by simplest fractions (i.e., sums of Cauchy kernels with unit coefficients). We present a solution of Chui's conjecture in the context of 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 equidistributed on the circle. We present sharp asymptotics of these norms. Finally, we describe the closure of the simplest fractions in weighted Bergman spaces under consideration. The talk is based on a joint work with E. Abakumov (Univ. Gustave Eiffel) and A. Borichev (Aix–Marseille Univ.).