Three Current Topics on Approximation by Analytic Functions

We are dealing with three current topics in the theory of approximation by analytic functions that attract significant interest of analysts during last two decades. The first one is the problem on uniform approximation of functions on compact sets in the plane by polynomial solutions of homogeneous second-order elliptic equations with constant complex coefficients and by systems of such equations. One of the most intriguing open questions concerning this problem is the question whether the analogue of the Walsh–Lebesgue criterion for uniform approximation by harmonic polynomials holds true in the case of approximation by polynomial solutions of general homogeneous second-order strongly elliptic equations with constant complex coefficients. The second topic encompasses several problems on uniform approximation of functions on compact sets in the plane by elements of polynomial and rational modules of polyanalytic type. We are dealing with approximation by functions of the form $$p_{1}g+p_{0}$$ , where $$p_{0}$$ and $$p_{1}$$ are polynomials or rational functions in the complex variable, while $$g$$ is some fixed function (the generator of the corresponding module) possessing certain natural regularity conditions. One interesting and important problem in this topic is to prove the analogue of the famous Verdera’s conjecture for modules with general generators. Finally, the third topic concerns approximation by simplest sums (that is, quantized sums with unit coefficients) of Cauchy kernels and other analytic kernels. In these three topics we present the studies state of the art and highlight some open problems awaiting for solution.