Baran O. Approximation of functions of multiple variables by branched con-tinued fractions with inequivalent variables.

In the thesis the branched continued fractions with inequivalent variables and the branched continued fractions of special form have been defined, their basic properties have been investigated (equivalence, difference formulae between two approximants), L.Lorentzen's results about properties of values sets and elements sets for continued fractions have been generalized. Correspondent and associated branched continued fractions with inequivalent variables to formal multiple power series have been constructed, these fractions have been used for interpolation of function of multiple variables, convergence conditions of this interpolation process have been set. For branched continued fractions with inequivalent variables and branched continued fractions of special form with complex elements multidimensional generalizations of continued fraction convergence criteria such as: Worpitsky, ?leshy?ski-Pringsheim ones, parabolic theorem of Thron have been established. Theorems (Leighton-Wall, Wyshinski-Mc Laughlin, Thron, Lange etc.) on twin convergence regions for continued fractions have been generalized.