Extension problem and bases for spaces of infinitely differentiable functions

Abstract

We examine the Mityagin problem: how to characterize the extension property in geometric terms. We start with three methods of extension for the spaces of Whitney functions. One of the methods was suggested by B. S. Mityagin: to extend individually the elements of a topological basis. For the spaces of Whitney functions on Cantor sets K( ), which were introduced by A. Goncharov, we construct topological bases. When the set K( ) has the extension property, we construct a linear continuous extension operator by means of suitable individual extensions of basis elements. Moreover, we use local Newton interpolations to contruct an extension operator. In the end, we show that for the spaces of Whitney functions, there is no complete characterization of the extension property in terms of Hausdorff measures or growth of Markov's factors.