APPROXIMATION BY INTERPOLATING POLYNOMIALS IN WEIGHTED SYMMETRIC SMIRNOV SPACES

Year 2012, Volume: 41 Issue: 5, 643 - 649, 01.05.2012

Ramazan Akgün Hüseyin Koç

Abstract

APPROXIMATION BY INTERPOLATING POLYNOMIALS IN WEIGHTED SYMMETRIC SMIRNOV SPACES

Abstract

Let Γ ⊂ C be a closed BR curve without cusps. In this work approximation by complex interpolating polynomials in a Weighted Symmetric Smirnov Space is studied. It is proved that the convergence rate of complex interpolating polynomials and the convergence rate of best approximating algebraic polynomials are the same in the norm of Symmetric Smirnov Spaces.

Keywords

Curve of bounded rotation, Faber polynomials, Interpolating polynomial, Symmetric Smirnov space, Cauchy singular operator

References

• Akg¨un, R. and Israfilov, D. M. Approximation by interpolating polynomials in Smirnov- Orlicz class, J. Korean Math. Soc. 43, 412–424, 2006.
• Benneth, C. and Sharpley, R. Interpolation of operators (Pure and Applied Mathematics 129, Academic Press, Boston, 1988).
• Boyd, D. W. Spaces between a pair of reflexive Lebesgue spaces, Proc. Amer. Math. Soc. 18, 215–219, 1967.
• Gaier, D. Lectures on complex approximation (Birkh¨auser, Boston, Basel, Stuttgart, 1987). [5] Garnett, J. B. Bounded analytic functions (Pure and Applied Mathematics 96, Academic Press Inc. [Harcourt Brace Jovanovich, Publishers], New York, London, 1981). [6] Karlovich,
• A. Yu. Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights, J. Operator Theory 47, 303– 323, 2002.
• H. Ko¸c, Convergence of interpolating polynomials in symmetric function spaces (M.Sci. Thesis, Balikesir University, Institute of Science, 2011).
• Pommerenke, Ch. Conforme abbildung und Fekete-punkte, Mathematische Zeitschrift 89, 422–438, 1965.
• Pommerenke, Ch. Boundary behaviour of conformal maps (Grundlehren der Mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 299, Springer- Verlag, Berlin, 1992).
• Shen, X. C. and Zhong, L. On Lagrange interpolation in Ep(D) for 1 < p < ∞, (Chinese) Adv. Math. 18, 342–345, 1989.
• Smirnov, V. I. and Lebedev, N. A. Functions of complex variable: Constructive theory (The M. I. T. Press, Cambridge, Mass., 1968).
• Suetin, P. K. Series of Faber Polynomials (Gordon and Breach, 1 Reading, 1998).
• Zhong, L. and Zhu, L. Convergence of the interpolants based on the roots of Faber polyno- mials, Acta Math. Hungarica 65 (3), 273–283, 1994.
• Zhu, L. Y. A new class of interpolation nodes, (in Chinese) Advances in Mathematics 24 (4), 327-ˆu334, 1995.