Description of a Class of Functions which Admit an Approximation by Rational Functions with Preassigned Poles I

Abstract

In classical complex function theory there are problems for scalar functions which can be better understood and studied in the framework of J-contractive analytic matrix functions. This point of view was initiated by V.P.Potapov who created the theory of J-contractive analytic matrix functions (originally with other motivations). Propagating this viewpoint he stressed the fact of the same nature that some problems of real analysis can be better understood by using methods of complex function theory. The validity of this point of view was confirmed by V.P.Potapov’s own work, and investigations of his followers, on extrapolation and interpolation problems for functions belonging to special classes (of I.Schur, C.Caratheodory, M.G.Krein, etc.) and on approximation problems for scalar functions. The present article, which is devoted to the solution of a problem for scalar analytic functions which was posed 25 years ago by G.Ts.Tumarkin, can be regarded as further evidence of the fruitfulness of the ideology of V.P.Potapov.