Approximation and Extension in Normed Spaces of Infinitely Differentiable Functions

We consider questions of rational and polynomial approximation, and related extension questions, for various normed spaces of infinitely differentiable functions on perfect compact subsets of the complex plane C and the real line R. We obtain an approximation theorem for compact planar sets which are, in a precise sense, locally radially– self–absorbing. All smoothly–bounded compact sets are of this type. We give a variety of results and counterexamples on extension, mainly in the one-dimensional case. We also prove polynomial approximation theorems for totally–disconnected sets, linear sets, and some others.