Browsing by Author "Goncharov, Alexander"

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    Bases in some spaces of Whitney functions
    (Duke University Press, 2017-06) Goncharov, Alexander; Ural, Zeliha
    We 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.
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    Bases in the spaces of Whitney jets
    (Birkhaeuser Science, 2022-01-10) Goncharov, Alexander
    We construct a basis in the space of Whitney jets defned on a null sequence in ℝ with a moderate rate of convergence.
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    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, Yasemin
    We 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).
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    On the absence of stability of bases in some Fréchet spaces
    (Springer Science and Business Media B.V., 2020-08) Goncharov, Alexander
    We show that, for each compact subset of the real line of infinite cardinality with an isolated point, the space of Whitney jets on the set does not possess a basis consisting only of polynomials. On the other hand, polynomials are dense in any Whitney space. Thus, there are no general results about stability of bases in Fréchet spaces.
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    On the geometric characterization of the extension property
    (The Belgian Mathematical Society, 2007) Goncharov, Alexander
    A geometric characterization of the extension property is given for Cantortype sets. The condition can also be done in terms of the rate of growth of certain sequences to the Robin constants of local parts of the set.
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    Quasi-equivalence of bases in some Whitney spaces
    (Cambridge University Press, 2021-05-18) Goncharov, Alexander; Şengül, Yasemin
    If 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.
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    Some asymptotics for extremal polynomials
    (Springer, 2016) Alpan, Gökalp; Goncharov, Alexander; Hatinoğlu, B.
    We review some asymptotics for Chebyshev polynomials and orthogonal polynomials. Our main interest is in the behaviour of Widom factors for the Chebyshev and the Hilbert norms on small sets such as generalized Julia sets.