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A compact set without Markov's property but with an extension operator for $C^∞$-functions

100%
Goncharov A.
Studia Mathematica
|
1996
|
tom 119
|
nr 1
27-35
EN
We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L: ℇ(K) → C^{∞}[0,1]$. At the same time, Markov's inequality is not satisfied for certain polynomials on K.
2
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Widom factors for the Hilbert norm

63%
Alpan G., Goncharov A.
Banach Center Publications
|
2015
|
tom 107
|
nr 1
11-18
EN
Given a probability measure μ with non-polar compact support K, we define the n-th Widom factor W²ₙ(μ) as the ratio of the Hilbert norm of the monic n-th orthogonal polynomial and the n-th power of the logarithmic capacity of K. If μ is regular in the Stahl-Totik sense then the sequence $(W²ₙ(μ))_{n=0}^{∞}$ has subexponential growth. For measures from the Szegő class on [-1,1] this sequence converges to some proper value. We calculate the corresponding limit for the measure that generates the Jacobi polynomials, analyze the behavior of the corresponding limit as a function of the parameters and review some other examples of measures when Widom factors can be evaluated.
3
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Smoothness of the Green function for a special domain

63%
Celik S., Goncharov A.
Annales Polonici Mathematici
|
2012
|
tom 106
|
nr 1
113-126
EN
We consider a compact set K ⊂ ℝ in the form of the union of a sequence of segments. By means of nearly Chebyshev polynomials for K, the modulus of continuity of the Green functions $g_{ℂ∖K}$ is estimated. Markov's constants of the corresponding set are evaluated.
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