Approximation by simplest sums

We will discuss the question on approximation by simplest fractions (i.e., sums of Cauchy kernels with unit coefficients). We will focus on 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. The sharp asymptotics of these norms will be presented. 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 with E. Abakumov (Univ. Gustave Eiffel, Paris, France) and A. Borichev (Aix–Marseille University, France). Also the problem will be touched on concerning approximation of functions by simplest bianalytic fractions that is sums with unit coefficients of fundamental solutions to the Bitsadze equation.