Extension of Functions with ω-Rapid Polynomial Approximation Original Research Article

For a weight function ω : [0, ∞[ → [0, ∞[ we denote by E(ω)(RN) the class of all ω-ultradifferentiable functions of Beurling type on RN. Each element in E(ω)(RN) is a function with ω-rapid polynomial approximation on each compact set K ⊂ of RN, whenever ω is a strong weight function, i.e., [formula] where PNj denotes the space of all polynomials in N variables of degree ≤ l and ∥ ∥K denotes the sup-norm on K. In the present pages there is given a family of weight functions ω such that each function f with ω-rapid polynomial approximation defined on a compact set K satisfying Markov′s inequality can be extended to an ω-ultradifferentiable function on RN. However this is not true for the small Gevrey classes E(t1/d) = Γ(d).