Chui’s conjecture in Bergman spaces

In the talk we present a solution of 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 equidistributed on the circle. We present sharp asymptotics of these norms. Finally, we describe the closure of the simplest fractions in weighted Bergman spaces.
The talk is based on a joint work with E. Abakumov (Univ. Gustave Eiffel) and A. Borichev (Aix–Marseille Univ.).