Polynomial Approximation on Varying Sets Original Research Article

Let {γm}m=1∞be a sequence of positive numbers, and letf:Rd→Cbe a function such that for someC=Cf<∞ and everyξ∈Rdthere exist polynomialsPm(x)=Pm(x; ξ), degPm≤m,m=0, 1, ..., satisfying inequalities[formula]In this paper the authors study smoothness, quasianalytic and analytic properties offin terms of the sequence {γm}m=1∞. The results are new even for the case thatPmare 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-functionϕ:Rd→R,d>1, that algebraic polynomials are dense inCϕ|A0(A) for every affine subspaceA⊂Rdof dimension less thand, but are not dense in the spaceCϕ0(Rd).