Approximation by bianalytic simplest fractions (sums of shifts of the function $\overline{z}/z$)

In the talk, based mainly on the recent joint work of the author and P. Borodin, it is planned to discuss the problem on approximation of functions on compact subsets of the complex plane by sums of shifts of the function $\overline{z}/z$ which is (up to the constant factor $1/\pi$) the fundamental solution to the Bitsadze partial differential operator (that determines bianalytic functions); such sums are known as bianalytic simplest fractions. This problem is recently appeared in connection with the topic on approximation of functions by quantized sums of various kernels, which is an actively studied subject in the current approximation theory. The problem under consideration is also motivated by recent progress in studies of approximation by usual simplest fractions (the sums of Cauchy kernels with unit coefficients). It is planned to present new results in the problem mentioned above and discuss new phenomena, appearing concerning the matter.