Polynomial Approximation for Analytic Continuation from a Boundary Arc

In this paper, we present a construction of polynomials approximating a function that is analytic in a domain and continuous on a fixed arc of the boundary, and besides, to construct the polynomials, we use only the values of the function on this arc. More precisely, we consider a bounded, simply connected domain in the complex plane with a chord-arc boundary. Furthermore, we assume that the function grows no faster than a power of the distance to the boundary as it approaches points on the boundary that do not lie on the fixed arc. At one of the stages of constructing the approximating polynomials, we use an analogue of the famous Carleman’s formula, which allows us to reconstruct the value of an analytic function from its values on a boundary arc.