Speaker
Description
We will discuss the question on approximation by simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) and by simplest bianalytic sums (i.e., sums of fundamental solutions to the Bitsadze equation with unit coefficients). We will start with Chui's conjecture and its version for weighted (Hilbert) Bergman spaces. 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. Next, we describe the closure of the simplest fractions in weighted Bergman spaces under consideration. These results were obtained at 2021 in the joint work by the speaker with E.~Abakumov (Univ. Gustave Eiffel, Paris, France) and A.~Borichev (Aix–Marseille University, France). Finally,
we discuss the problem on approximation of functions by simplest bianalytic fractions,and several new effects and phenomena that appeared in this connection. This part is based on the joint work in progress by the speaker with P.~Borodin (Lomonosov Moscow State University).
