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On Bernstein–Walsh-type lemmas in regions of the complex plane

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Ukrainian Mathematical Journal Aims and scope

Let \( G \subset {\mathbb C} \) be a finite region bounded by a Jordan curve \( L: = \partial G \), let \( \Omega : = {\text{ext}}\bar{G} \) (with respect to \( {\overline {\mathbb C}} \)), \( \Delta : = \left\{ {z:\left| z \right| > 1} \right\} \), and let \( w = \Phi (z) \) be a univalent conformal mapping of Ω onto Δ normalized by \( \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 \). By A p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

$$ \left\| f \right\|_{Ap}^p(G): = \int\limits_G {{{\left| {f(z)} \right|}^p}d{\sigma_z} < \infty, } $$
(*)

where σ is a two-dimensional Lebesgue measure. Let P n (z) be arbitrary algebraic polynomial of degree at most n: The well-known Bernstein–Walsh lemma says that

$$ \left\| {{P_n}(z)} \right\| \leq {\left| {\Phi (z)} \right|^{n + 1}}{\left\| {{P_n}} \right\|_{C\left( {\bar{G}} \right)}},\quad z \in \Omega . $$
(**)

First, we study the problem of estimation (**) for the norm (*). Second, we continue studying estimation (**) by replacing the norm \( {\left\| {{P_n}} \right\|_{C\left( {\bar{G}} \right)}} \) with \( {\left\| {{P_n}} \right\|_{{A_2}(G)}} \) for some regions of the complex plane.

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References

  1. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, RI (1960).

    MATH  Google Scholar 

  2. E. Hille, G. Szegö, and J. D. Tamarkin, “On some generalization of a theorem of A. Markoff,” Duke Math. J., 3, 729–739 (1937).

    Article  MathSciNet  Google Scholar 

  3. O. Lehto and K. I. Virtanen, Quasiconformal Mapping in the Plane, Springer, Berlin (1973).

    Google Scholar 

  4. S. Rickman, “Characterisation of quasiconformal arcs,” Ann. Acad. Sci. Fenn., Ser. A. Math., 395, 1–30 (1966).

    MathSciNet  Google Scholar 

  5. F. G. Abdullayev, “On some properties of orthogonal polynomials over an area in domains of the complex plane. III,” Ukr. Math. J., 53, No. 12, 1934–1948 (2001).

    Article  Google Scholar 

  6. F. G. Abdullayev, “The properties of the orthogonal polynomials with weight having singularity on the boundary contour,” J. Comp. Anal. Appl., 6, No. 1, 43–59 (2004).

    MathSciNet  MATH  Google Scholar 

  7. V. V. Andrievskii, V. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions on Complex Plane, World Federation, Atlanta (1995).

    Google Scholar 

  8. L. V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand, Princeton, NJ (1966).

    MATH  Google Scholar 

  9. N. Stylianopoulos, “Fine asymptotics for Bergman orthogonal polynomials over domains with corners,” CMFT, 2009, Ankara (2009).

  10. V. V. Andrievskii, “Constructive characterization of harmonic functions in domains with quasiconformal boundary,” in: Quasiconformal Continuation and Approximation by Functions in a Set of the Complex Plane [in Russian], Kiev (1985).

  11. F. G. Abdullayev, Ph.D. Thesis, Donetsk (1986).

  12. F. G. Abdullayev, “Uniform convergence of generalized Bieberbach polynomials in regions with nonzero angles,” Acta Math. Hung., 77, No. 3, 223–246 (1997).

    Article  MathSciNet  MATH  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 3, pp. 291–302, March, 2011.

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Abdullayev, F.G., Aral, N. On Bernstein–Walsh-type lemmas in regions of the complex plane. Ukr Math J 63, 337–350 (2011). https://doi.org/10.1007/s11253-011-0507-y

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  • DOI: https://doi.org/10.1007/s11253-011-0507-y

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