On approximation to functions satisfying a generalized continuity condition

Abstract

study of approximation to functions by sequences of polynomials is divided into Problems a and f. In Problem a we investigate the relation between continuity properties on C of the function and degree of convergence on Z of approximating polynomials; in Problem ,B we investigate the relation between continuity properties on CR of the function and degree of convergence on C of the approximating polynomials. For each of these problems, results obtained are of two types: direct theorems and indirect theorems. In a direct theorem given certain continuity properties of a function, we establish the existence and degree of convergence of a sequence of approximating polynomials; in an indirect theorem given a sequence of polynomials converging to a function with a certain degree, we establish continuity properties of the function. We shall understand by a harmonic function or harmonic polynomial a real harmonic function or polynomial. The letter s will denote arc-length measured along the curve in question; akU(z)/OSk will indicate the kth derivative of u(z) with respect to arc-length s. For any function F(z), FM (z) F(z). The letter k will be reserved for positive integers and zero. The letters L and M with or without subscripts denote positive constants which may vary from one theorem and its proof to another and may depend on the point sets involved,