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 ofinfinitely differentiable functions on perfect compact subsetsof the complex plane C and the real line R. We obtain an approximationtheorem for compact planar sets which are, in a precise sense,locally radially self-absorbing. All smoothly-bounded compactsets are of this type. We give a variety of results and counterexampleson extension, mainly in the one-dimensional case. We also provepolynomial approximation theorems for totally-disconnected sets,linear sets, and some others.