Near best approximation property of interpolation and Poisson polynomials in weighted variable exponent Smirnov classes

Abstract

Let $G$ be a bounded Jordan domain in the complex plane $\mathbb{C}$. In this work under some restrictions of ${G}$ the near best approximation property of complex interpolation and Poisson polynomials based on the Faber polynomials of $\overline{{G}}$ in the weighted variable exponent Smirnov classes ${E}_{\omega }^{p(\cdot )}{(G)}$ are proved.

Keywords

• interpolating polynomials
• Faber polynomials
• weighted variable exponent Smirnov classes
• Poisson polynomials
• near the best approximation

References

1. [1] R. Akgun and D.M. Israfilov, Approximation by interpolating polynomials in Smirnov Orlicz class, J. Korean Math. Soc. 43, 412-424, 2006.
2. [2] R. Akgun, Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces, Journal of Function Spaces and Applications 2012, Article ID 982360, 41 pages http://dx.doi.org/ 10.1155 /2012/982360.
3. [3] R. Akgun and H. Koc, Approximation by interpolating polynomials in weighted symmetric Smirnov spaces, Hacettepe Journal of Mathematics and Statistics 41 (5), 643- 649, 2012.
4. [4] D.V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces Foundation and Harmonic Analysis, Birkhäsuser, 2013.
5. [5] D.V. Cruz Uribe and D.L. Wang, Extrapolation and weighted norm inequalities in the variable Lebesgue spaces, Trans. Am. Math. Soc. 369 (2), 1205–1235, 2017.
6. [6] D. Gaier, The Faber operator and its boundedness, J. Approx. Theory 101 (2), 265- 277, 1999.
7. [7] D. Gaier, Lectures on complex approximation, Birkhäuser, 1987.
8. [8] G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translation of Mathematical Monographs AMS 26, 1969.

9. [9] G.David, Operateurs integraux singulers sur certaines courbes du plan complexe, Ann. Sci. Ecole Norm. Sup. 4 (17), 157–189, 1984.
10. [10] L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with Variable exponents, Springer, 2017.
11. [11] E.M. Dynkin and B.P. Osilenker, Weighted estimates for singular integrals and their applications, In: Mathematical Analysis 21, Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 42–129, 1983.
12. [12] P.L. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York, 1970.
13. [13] I.I Ibragimov and D.I. Mamedkhanov, A constructive characterization of a certain class of functions, Dokl. Akad. Nauk SSSR 223, 35-37, 1975, Soviet Math. Dokl. 4, 820-823, 1976.
14. [14] D.M. Israfilov, Approximation by $p$− Faber polynomials in the weighted Smirnov class $E^{p}\left( G,\omega\right) $ and the Bieberbach polynomials, Constr. Approx. 17, 335-351, 2001.
15. [15] D.M. Israfilov, Approximation by p-Faber-Laurent Rational functions in weighted Lebesgue spaces, Czechoslovak Mathematical Journal 54 (129), 751-765, 2004.
16. [16] D.M. Israfilov and A. Guven, Approximation in weighted Smirnov Classes. East Journal on Approximation 11 (1), 91-102, 2005.
17. [17] D.M. Israfilov and A. Testici, Approximation in weighted Smirnov spaces, Complex variable and elliptic equations 60 (1), 45-58, 2015.
18. [18] D.M. Israfilov and A. Testici, Approximation in Smirnov classes with variable exponent ,Complex variable and elliptic equations 60 (9) ,1243-1253, 2015.
19. [19] D.M. Israfilov and A. Testici, Multiplier and Approximation Theorems in Smirnov Classes with Variable Exponent, Turkish Journal of Mathematics 42, 1442-1456, 2018.
20. [20] D.M. Israfilov and A. Testici, Approximation by Faber-Laurent rational functions in Lebesgue spaces with variable exponent, Indagationes Mathematicae 27 (4), 914-922, 2016.
21. [21] D.M. Israfilov and A. Testici, Some inverse and simultaneous approximation theorems in weighted variable exponent Lebesgue spaces, Analysis Mathematica 44, 475–492, 2018.
22. [22] S.Z. Jafarov, Approximation by Rational Functions in Smirnov-Orlicz Classes, Journal of Mathematical Analysis and Applications 379, 870-877, 2011.
23. [23] S.Z. Jafarov, Approximation of functions by p -Faber – Laurent rational functions, Complex Variables and Elliptic Equations 60 (3), 416-428, 2015.
24. [24] V. Kokilashvili, A direct theorem on Mean Approximation of Analytic Functions by Polynomials, Soviet Math. Dokl. 10, 411-414, 1969.
25. [25] V. Kokilashvili and S. Samko, Operators of harmonic analysis in weighted spaces with non-standard growth, J. Math. Anal. Appl. 352, 15–34, 2009.
26. [26] C. Pommerenke, Conforme abbildung und Fekete-punkte, Math. Z. 89, 422-438, 1965.
27. [27] X.C. Schen and L. Zhong, On Lagrange Interpolatin in $E^{p}\left( D\right) $ for $1
28. [28] I.I. Sharapudinov, Some questions of approximation theory in the Lebesgue spaces with variable exponent, Itogi Nauki. Yug Rossii. Mat. Monograf. 5, Southern Mathematical Institute of the Vladikavkaz Scientic Center of the Russian Academy of Sciences and Republic of North Ossetia-Alania, 2012.
29. [29] I.I Sharapudinov, Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vall´ee-Poussin means, Sbornik: Mathematics 207(7), 1010–1036, 2016.
30. [30] A.I. Shvai, Approximation on analytic functions by Poisson polynomials, Ukrain. Math. J., 25 (6), 710-713, 1973.
31. [31] V.I. Smirnov and N.S. Lebedev, Functions of a Complex Variable, Constructive Theory: M.I.T. Press, 1968.
32. [32] P.K. Suetin, Series of Faber Polynomials, New York: Gordon and Breach Science Publishers, 1998.
33. [33] A.Testici, Some theorems of approximation theory in weighted Smirnov classes with variable exponent, Computational Methods and Function Theory 20, 39–61, 2020.
34. [34] A. Testici, Approximation theorems in weighted Lebesgue spaces with variable exponent, Filomat 35 (2), 561–577, 2021.
35. [35] L. Zhong and L. Zhu, Convergence of the interpolants based on the roots of Faber polynomials, Acta Math. Hungar. 65 (3), 273-283, 1994.
36. [36] L.Y. Zhu, A kind of interpolation nodes, (Chinese), Adv. Math., 1994.