Approximation by sums of shifts of the function $\overline{z}/z$

P. A. Borodin^ab, K. Yu. Fedorovskiy<sup>ab

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b St. Petersburg State University, St. Petersburg, Russia

Abstract: Uniform approximations by sums of bianalytic kernels, that is, sums of shifts of the function $\overline{z}/z$ are under consideration. Namely we study conditions on a domain $\Omega$ in the complex plane $\mathbb{C}$ and a set $E\subset \mathbb{C}\setminus \Omega$ ensuring that any bianalytic function in $\Omega$ can arbitrarily well be approximated locally uniformly inside $\Omega$ by sums of bianalytic kernels with singularities on $E$. Also conditions on a compact set $X\subset\mathbb{C}$ are investigated ensuring that any continuous function on $X$ that is bianalytic in the interior of $X$ can arbitrarily well be uniformly approximated on $X$ by sums of bianalytic kernels with singularities in $\mathbb{C}\setminus X$.
In both cases the necessary or sufficient conditions bound are significantly different from the corresponding result on approximations by simple partial fractions, that is, sums of shifts of the function $1/z$.

Keywords: uniform approximation, sums of shifts, bianalytic function, simple partial fraction, Nevanlinna domain, Carathéodory compact set.

Received: 28.02.2025 and 27.08.2025

DOI: 10.4213/sm10298