Approximation by simplest fractions and simplest bianalytic fractions

This talk will examine issues of uniform approximation of functions in domains and on compact sets in the complex plane by sums of analytic kernels (which can be fundamental solutions of various elliptic differential equations, reproducing kernels of various spaces of analytic functions, and other similar analytic objects). Importantly, the approximation aggregates are sums with unit coefficients (such sums are often called simplest or quantized ones). Particular attention will be paid to approximation by simplest fractions (sums of Cauchy kernels) and approximation by simplest bianalytic fractions (sums of bianalytic kernels-fundamental solutions to the Bitsadze equation). Recent results and open problems will be presented.