Browsing by Subject "Topological bases"
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Item Open Access Bases in Banach spaces of smooth functions on Cantor-type sets(Elsevier, 2011) Goncharov, A. P.; Ozfidan, N.We suggest a Schauder basis in Banach spaces of smooth functions and traces of smooth functions on Cantor-type sets. In the construction, local Taylor expansions of functions are used. © 2011 Elsevier Inc.Item Open Access Bases in some spaces of Whitney functions(Duke University Press, 2017-06) Goncharov, Alexander; Ural, ZelihaWe construct topological bases in spaces of Whitney functions on Cantor sets, which were introduced by the first author. By means of suitable individual extensions of basis elements, we construct a linear continuous exten- sion operator, when it exists for the corresponding space. In general, elements of the basis are restrictions of polynomials to certain subsets. In the case of small sets, we can present strict polynomial bases as well.Item Open Access Bases in the spaces of Whitney jets(Birkhaeuser Science, 2022-01-10) Goncharov, AlexanderWe construct a basis in the space of Whitney jets defned on a null sequence in ℝ with a moderate rate of convergence.Item Open Access Extension problem and bases for spaces of infinitely differentiable functions(2017-04) Merpez, Zeliha UralWe 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.Item Open Access Logarithmic dimension and bases in Whitney spaces(Scientific and Technical Research Council of Turkey - TUBITAK,Turkiye Bilimsel ve Teknik Arastirma Kurumu, 2021-07-27) Goncharov, Alexander; Şengül Tezel, YaseminWe give a formula for the logarithmic dimension of the generalized Cantor-type set K . In the case when the logarithmic dimension of K is smaller than 1, we construct a Faber basis in the space of Whitney functions E(K).Item Open Access Logarithmic dimension and bases in whitney spaces(2006) Şengül, YaseminIn generalization of [3] we will give the formula for the logarithmic dimension of any Cantor-type set. We will demonstrate some applications of the logarithmic dimension in Potential Theory. We will construct a polynomial basis in E(K(Λ)) when the logarithmic dimension of a Cantor-type set is smaller than 1. We will show that for any generalized Cantor-type set K(Λ), the space E(K(Λ)) possesses a Schauder basis. Locally elements of the basis are polynomials. The result generalizes theorems 1 and 2 in [12].Item Open Access Quasi-equivalence of bases in some Whitney spaces(Cambridge University Press, 2021-05-18) Goncharov, Alexander; Şengül, YaseminIf the logarithmic dimension of a Cantor-type set K is smaller than 1 , then the Whitney space E(K) possesses an interpolating Faber basis. For any generalized Cantor-type set K, a basis in E(K) can be presented by means of functions that are polynomials locally. This gives a plenty of bases in each space E(K) . We show that these bases are quasi-equivalent.