Browsing by Subject "Extension operator"

Now showing 1 - 3 of 3

Open Access
Extension operators for spaces of infinitely differentiable functions
(2005) Altun, Muhammed
We start with a review of known linear continuous extension operators for the spaces of Whitney functions. The most general approach belongs to PawÃlucki and Ple´sniak. Their operator is continuous provided that the compact set, where the functions are defined, has Markov property. In this work, we examine some model compact sets having no Markov property, but where a linear continuous extension operator exists for the space of Whitney functions given on these sets. Using local interpolation of Whitney functions we can generalize the PawÃlucki-Ple´sniak extension operator. We also give an upper bound for the Green function of domains complementary to generalized Cantor-type sets, where the Green function does not have the H¨older continuity property. And, for spaces of Whitney functions given on multidimensional Cantor-type sets, we give the conditions for the existence and non-existence of a linear continuous extension operator.
Open Access
Extension problem and bases for spaces of infinitely differentiable functions
(2017-04) Merpez, Zeliha Ural
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.
Open Access
A local version of the Pawlucki-Plesniak extension operator
(Elsevier, 2005-01) Altun, M.; Goncharov, A.
Using local interpolation of Whitney functions, we generalize the Pawłucki and Pleśniak approach to construct a continuous linear extension operator. We show the continuity of the modified operator in the case of generalized Cantor-type sets without Markov's Property.