Rational approximation of analytic functions: potential theory approach of Gonchar and modern development

We recall the main concepts proposed by A. A. Gonchar for determination of the degree of approximation of analytical functions by rational functions. Namely asymptotical reduction of the best rational approximants to the multipoint Padé approximants and orthogonal polynomials with respect to a “varying” weight. Then we consider modern development of these concepts related to the strong asymptotics for polynomials orthogonal on the extremal compacts of the complex plane with respect to complex varying weights.