Some problems of approximation theory and Rubel's interpolation problem

Let E be a compact plane set with connected complement, and let A(E) be the class of all complex continuous function on E that are analytic in the interior of E. Let the function f from A(E) be zero free in the interior of E. By Mergelyan’s theorem f can be uniformly approximated on E by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on E? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets E. The present talk describes classes of functions for which the zero free approximation is possible on an arbitrary E. Another polynomial approximation problem proposed by L. Zalcman in 1982 will be solved in this talk.  Also, the solution to an interpolation problem in the unit disc proposed by L. Rubel in 1973 will be presented.