Polynomial approximation on varying sets

Let {gamma(m)}(alpha)(m = 1) be a sequence of positive numbers, and let f: R(d) --> C be a function such that for some C = C-f < infinity and every xi is an element of R(d) there exist polynomials P-m(x) = P-m(x; xi), deg P-m less than or equal to m, m = 0, 1,..., satisfying inequalities sup{1f(x) - P-m(x; xi)\:\x - xi\ less than or equal to gamma(m)} less than or equal to C exp{-m}. In this paper the authors study smoothness, quasianalytic and analytic properties of f in terms of the sequence {gamma(m)}(alpha)(m = 1). The results are new even for the case that P-m are Taylor polynomials. Using them, the authors prove a Cartwright-type theorem on entire functions of exponential type bounded on some discrete subset of the real hyperplane and construct such a weight-function phi: R(d) --> R, d > 1, that algebraic polynomials are dense in C-phi iA(0) (A) for every affine subspace A subset of R(d) of dimension less than d, but are not dense in the space C-phi(0)(R(d)). (C) 1996 Academic Press, Inc.