Bases in the spaces of infinitely differentiable and Whitney functions.
by
A. Goncharov
Bilkent University, Ankara, Turkey
Coauthors: V.P. Zahariuta (Rostov State University, Russia and TUBITAK, Turkey)

First we construct a basis in the space E(K) of Whitney functions for the compact set K being a sequence of closed intervals tending to a point. The construction is based on a convolution property for the coefficients of scaling Chebyshev's polynomials. The method works under some restrictions on K, but it can be applied in two main cases:

1) when the space E(K) is isomorphic to the space s of rapidly decreasing sequences ( here using this basis we construct a special basis in the space C^\infty[0, 1] ) and

2) when the lengths of the intervals in K tend to zero arbitrarily quickly ( here we have a continuum of pairwise nonisomorphic spaces).

Next we use the special basis in the space C^\infty[0, 1] to construct a basis in the space of infinitely differentiable functions on a graduated sharp cusp with arbitrary sharpness of the cusp.

Extending the basis elements we give an explicit form of a continuous linear extension operator for both cases provided that such an operator exists. The method generalizes Mitiagin's construction of the extension operator for K=[-1, 1]. Comparing this approach with the more general technique of Pawlucki and Plesniak who suggested the extension operator in the form of a series containing Lagrange interpolation polynomials with Fekete-Leja system of knots under the hypothesis of Markov's Property of a compact set , we present an example of a compact set K without the Markov Property and such that the corresponding extension operator can be given by means of extensions of the basis elements of the space E(K).

At last we discuss the method to construct a basis in the space of Whitney functions defined on the Cantor ternary set.

(T)

Date received: December 16, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cado-77.