Let the integer and the modulus of continuity be fixed, and let be the class of all functions continuous on the closed unit disk, analytic on its interior, and having an -continuous th derivative on. Consider for each and each fixed the polynomial in (the st partial sum of the Taylor series of in a neighborhood of). Then for any two points (1.1)Let be a closed subset of. This article contains a solution of the problem of free interpolation in, formulated as follows: Find necessary and sufficient conditions on such that for each collection of th-degree polynomials satisfying conditions of the type (1.1) for all there is a function with. Bibliography: 13 titles.