Approximation by $p-$Faber-Laurent rational functions in doubly-connected domain

Year 2019, Volume: 48 Issue: 5, 1356 - 1366, 08.10.2019

Sadulla Z. Jafarov

Abstract

Let $G$ be a doubly-connected domain bounded by regular curves. In this work, the approximation properties of the $p-$Faber-Laurent rational seriesexpansions in the $\omega -$weighted Smirnov classes $E^{p}(G,\omega )$ are studied.

Keywords

Faber-Laurent rational functions, conformal mapping, regular curve, $\omega$-weighted Smirnov class $E^{p}(G;\omega )$, $k$-th integral modulus of continuity

References

• [1] S.Y. Alper, Approximation in the mean of analytic functions of class $E^{p}$ (in Russian), in: Investigations on the Modern Problems of the Function Theory of a Complex Variable, Gos. Izdat. Fiz.-Mat. 272-2386, Lit. Moscow, 1960.
• [2] J.E. Andersson, On the degree of polynomial approximation in $E^{p}(D)$, J. Approx. Theory 19, 61-68, 1977.
• [3] A. Cavus and D.M. Israfilov, Approximation by Faber-Laurent retional functions in the mean of functions of the class $L_{p}(\Gamma )$ with $1<p<\infty$, Approx. Theory Appl. 11 (1), 105-118, 1995.
• [4] P.L. Duren, Theory of $H^{p}$ spaces, Academic Press, 1970.
• [5] E.M. Dyn’kin, The rate of polynomial approximation in complex domain, in: Complex Analysis and Spectral Theory, 90-142, Springer-Verlag, Berlin, 1980.
• [6] E.M. Dyn’kin and B.P. Osilenker, Weighted estimates for singular integrals and their appllications, in: Mathematical Analysis 21., 42-129, Akad. Nauk. SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983.
• [7] G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translation of Mathematical Monographs 26, Providence, RI: AMS, 1968.
• [8] A. Guven and D.M. Israfilov, Approximation in rearrangement invariant spaces on Carleson curves, East J. Approx. 12 (4), 381-395, 2006.
• [9] A. Guven and D.M. Israfilov, Improved inverse theorems in weighted Lebesgue and Smirnov spaces, Bul. Belg. Math. Soc. Simon Stevin 14, 681-692, 2007.
• [10] E.A. Haciyeva, Investigation of the properties of functions with quasimonotone Fourier coefficients in generalized Nikolsky- Besov spaces, Author’s summary of can- didates dissertation, Tbilisi, (In Russian).
• [11] I.I. Ibragimov and J.I. Mamedkanov, A constructive characterization of a certain class of functions, Dokl. Akad. Nauk SSSR 223, 35-37, 1975, Soviet Math. Dokl. 4, 820-823, 1976.
• [12] D.M. Israfilov, Approximate properties of the generalized Faber series in an integral metric (in Russian), Izv. Akad. Nauk Az. SSR, Fiz.-Tekh. Math. Nauk 2, 10-14, 1987.
• [13] D.M. Israfilov, Approximation by p-Faber polynomials in the weighted Smirnov class $E^{p}(G,w)$ and the Bieberbach polynomials, Constr. Approx. 17, 335-351, 2001.
• [14] D.M. Israfilov, Approximation by p−Faber-Laurent rational functions in weighted Lebesgue spaces, Chechoslovak Math. J. 54, 751-765, 2004.
• [15] D.M. Israfilov and R. Akgün, Approximation by polynomials and rational functions in weighted rearrangement invariant spaces, J. Math.Anal. Appl. 346, 489-500, 2008.
• [16] D.M. Israfilov and A. Guven, Approximation in weighted Smirnov classes, East J. Approx. 11 (1), 91-102, 2005.
• [17] D.M. Israfilov and A. Testici, Improved converse theorems in weighted Smirnov spaces, Proc. Inst. Math. Mech. Natl.Acad. Sci. Azerb. 40 (1), 44-54, 2014.
• [18] D.M. Israfilov and A. Testici, Approximation by Faber-Laurent rational functions in Lebesgue spaces with variable exponent, Indagationes Math. 27, 914-922, 2016.
• [19] S.Z. Jafarov, On approximation of functions by p−Faber-Laurent rational functions, Complex Var. Elliptic Equ. 60 (3), 416-428, 2015.
• [20] X. Ji-Feng and Z. Ming, A complex approximation on doubly-connected domain, J. Fudan Univ. Nat. Sci. 44 (2), 328-331, 2005.
• [21] G.S. Kocharyan, On a generalization of the Laurent and Fourier series, Izv. Akad. Nauk Arm. SSr Ser, Fiz.-Mat. Nauk 11 (1), 3014, 1958.
• [22] V.M. Kokilashvili, On approximation of analytic functions from }$E_{p}$ classes (in Rus- sian), Trudy Tbiliss. Mat. Inst. im Razmadze Akad. Nauk Gruzin SSR 34, 82-102, 1968.
• [23] A.I. Markushevich, Analytic Function Theory: Vols. I, II, Nauka, Moscow, 1967.
• [24] P.K. Suetin, Series of Faber polynomials, Gordon and Breach Science Publishers, 1998.
• [25] H. Tietz, Faber series and the Laurent decomposition, Michigan Math. J. 4 (2), 157- 179, 1957.
• [26] H. Yurt and A. Guven, On rational approximation of functions in rearrangement invariant spaces, J. Class. Anal. 3 (1), 69-83, 2013.
• [27] H. Yurt and A. Guven, Approximation by Faber-Laurent rational functions on doubly connected domains, New Zealand J. Math. 44, 113-124, 2014.