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### Browsing Department of Mathematics by Author "Alpan, G."

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Item Open Access Asymptotic properties of Jacobi matrices for a family of fractal measures(Taylor and Francis, 2018) Alpan, G.; Goncharov, A.; Şimşek, A. N.Show more We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined, and different aspects of orthogonal polynomials on them were studied recently. Our main aim is to numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on (Formula presented.).Show more Item Open Access Chebyshev polynomials on generalized Julia sets(Springer, 2016) Alpan, G.Show more Let (fn)n=1∞ be a sequence of non-linear polynomials satisfying some mild conditions. Furthermore, let Fm(z) : = (fm∘ fm - 1⋯ ∘ f1) (z) and ρm be the leading coefficient of Fm. It is shown that on the Julia set J(fn), the Chebyshev polynomial of degree deg Fm is of the form Fm(z) / ρm- τm for all m∈ N where τm∈ C. This generalizes the result obtained for autonomous Julia sets in Kamo and Borodin (Mosc. Univ. Math. Bull. 49:44–45, 1994). © 2015, Springer-Verlag Berlin Heidelberg.Show more Item Open Access Orthogonal polynomials for the weakly equilibrium cantor sets(American Mathematical Society, 2016) Alpan, G.; Goncharov, A.Show more Let K(γ) be the weakly equilibrium Cantor-type set introduced by the second author in an earlier work. It is proven that the monic orthogonal polynomials Q2 s with respect to the equilibrium measure of K(γ) coincide with the Chebyshev polynomials of the set. Procedures are suggested to find Qn of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below. © 2016 American Mathematical Society.Show more Item Open Access Orthogonal Polynomials on Generalized Julia Sets(Birkhauser Verlag AG, 2017) Alpan, G.; Goncharov, A.Show more We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green’s functions and Parreau–Widom criterion for a special family of real generalized Julia sets. © 2017, Springer International Publishing.Show more Item Open Access Some open problems concerning orthogonal polynomials on fractals and related questions(Padova University Press, 2017) Alpan, G.; Goncharov, A.Show more We discuss several open problems related to analysis on fractals: Estimates of the Green functions, the growth rates of the Markov factors with respect to the extension property of compact sets, asymptotics of orthogonal polynomials and the Parreau-Widom condition, Hausdorff measures and the Hausdorff dimension of the equilibrium measure on generalized Julia sets. © 2017, Padova University Press. All rights reserved.Show more Item Open Access Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings(Springer Netherlands, 2016) Alpan, G.Show more Let μ be a probability measure with an infinite compact support on R. Let us further assume that Fn: = fn∘ ⋯ ∘ f1 is a sequence of orthogonal polynomials for μ where (fn)n=1 ∞ is a sequence of nonlinear polynomials. We prove that if there is an s0∈ N such that 0 is a root of fn′ for each n> s0 then the distance between any two zeros of an orthogonal polynomial for μ of a given degree greater than 1 has a lower bound in terms of the distance between the set of critical points and the set of zeros of some Fk. Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures. © 2016, Akadémiai Kiadó, Budapest, Hungary.Show more Item Open Access Two measures on Cantor sets(Elsevier, 2014-10) Alpan, G.; Goncharov, A.Show more We give an example of Cantor-type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in the Stahl–Totik sense. ⃝c 2014 Elsevier Inc. All rights reserved.Show more Item Open Access Widow factors for the Hilbert norm(Polska Akademia Nauk, 2015) Alpan, G.; Goncharov, A.Show more 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.Show more