Browsing by Subject "Chebyshev numbers"

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    Widom factors
    (Springer Netherlands, 2015) Goncharov, A.; Hatinoğlu, B.
    Given a non-polar compact set K,we define the n-th Widom factor Wn(K) as the ratio of the sup-norm of the n-th Chebyshev polynomial on K to the n-th degree of its logarithmic capacity. By G. Szegő, the sequence (Formula presented.) has subexponential growth. Our aim is to consider compact sets with maximal growth of the Widom factors. We show that for each sequence (Formula presented.) of subexponential growth there is a Cantor-type set whose Widom’s factors exceed Mn. We also present a set K with highly irregular behavior of the Widom factors.
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    Widom Factors
    (2014) Hatinoğlu, Burak
    In this thesis we recall classical results on Chebyshev polynomials and logarithmic capacity. Given a non-polar compact set K, we define the n-th Widom factor Wn(K) as the ratio of the sup-norm of the n-th Chebyshev polynomial on K to the n-th degree of its logarithmic capacity. We consider results on estimations of Widom factors. By means of weakly equilibrium Cantor-type sets, K(γ), we prove new results on behavior of the sequence (Wn(K))∞ n=1.By K. Schiefermayr[1], Wn(K) ≥ 2 for any non-polar compact K ⊂ R. We prove that the theoretical lower bound 2 for compact sets on the real line can be achieved by W2s (K(γ)) as fast as we wish. By G. Szeg˝o[2], rate of the sequence (Wn(K))∞ n=1 is slower than exponential growth. We show that there are sets with unbounded (Wn(K))∞ n=1 and moreoverfor each sequence (Mn)∞ n=1 of subexponential growth there is a Cantor-type set which Widom factors exceed Mn for infinitely many n. By N.I. Achieser[3][4], limit of the sequence (Wn(K))∞ n=1 does not exist in the case K consists of two disjoint intervals. In general the sequence (Wn(K))∞ n=1 may behave highly irregular. We illustrate this behavior by constructing a Cantor-type set K such that one subsequence of (Wn(K))∞ n=1 converges as fast as we wish to the theoretical lower bound 2, whereas another subsequence exceeds any sequence (Mn)∞ n=1 of subexponential growth given beforehand