Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes

  • A. Testici Balikesir Univ., Turkey
Keywords: Doubly connected domai, modulus of smoothnes, Faber-Laurent serie, generalized grand Smirnov clas, Carleson curve

Abstract

UDC 517.5

Let $G\subset \mathbb{C}$ be a doubly connected domain bounded by two rectifiable Carleson curves. In this work, we use the higher modulus of smoothness in order to investigate the approximation properties of $(p-\varepsilon)$-Faber–Laurent rational functions in the subclass of weighted generalized grand Smirnov classes ${E}^{p),\theta } ( {G,\omega })$ of analytic functions.

References

A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces, Collect. Math., 51, № 2, 131 – 148 (2000).

A. I. Markushevich, Analytic function Theory, vols I and II, Nauka, Moscow (1968).

A. Kinj, M. Ali, S. Mahmoud, Approximation by rational functions in Smirnov classes with variable exponent, Arab. J. Math., 37, № 1, 79 – 86 (2017), https://doi.org/10.1007/s40065-017-0164-6 DOI: https://doi.org/10.1007/s40065-017-0164-6

D. Gaier, Lectures on complex approximation, Boston-Stuttgart; Birkh¨auser-Verlag (1987). DOI: https://doi.org/10.1007/978-1-4612-4814-9

D. M. Israfilov, Approximation by $p$-Faber polynomials in the weighted Smirnov class $E^p(G, omega )$ and the Bieberbach polynomials, Constr. Approx., 17, 335 – 351 (2001), https://doi.org/10.1007/s003650010030 DOI: https://doi.org/10.1007/s003650010030

D. M. Israfilov, Approximation by $p$-Faber – Laurent Rational functions in weighted Lebesgue spaces, Czechoslovak Math. J., 54, 751 – 765 (2004), https://doi.org/10.1007/s10587-004-6423-7 DOI: https://doi.org/10.1007/s10587-004-6423-7

D. M. Israfilov, A. Guven, Approximation in weighted Smirnov Classes, East J. Approx., 11, 91 – 102 (2005). DOI: https://doi.org/10.3336/gm.40.1.09

D. M. Israfilov, A. Testici, Approximation in weighted Smirnov classes, Complex Variable and Elliptic Equat., 60, № 1, 45 – 58 (2015), https://doi.org/10.1080/17476933.2014.882915 DOI: https://doi.org/10.1080/17476933.2014.882915

D. M. Israfilov, A. Testici, Improved converse theorems in weighted Smirnov classes, Proc. Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, 40, № 1, 44 – 54 (2014).

D. M. Israfilov, A. Testici, Approximation in Smirnov classes with variable exponent, Complex Variable and Elliptic Equat., 60, № 9, 1243 – 1253 (2015), https://doi.org/10.1080/17476933.2015.1004539 DOI: https://doi.org/10.1080/17476933.2015.1004539

D. M. Israfilov, A. Testici, Approximation by Faber – Laurent rational functions in Lebesgue spaces with variable exponent, Indag. Math., 27, № 4, 914 – 922 (2016), https://doi.org/10.1016/j.indag.2016.06.001 DOI: https://doi.org/10.1016/j.indag.2016.06.001

D. M. Israfilov, A. Testici, Multiplier and approximation theorems in Smirnov classes with variable exponent, Turkish J. Math., 42, 1442 – 1456 (2018), https://doi.org/10.3906/mat-1707-15 DOI: https://doi.org/10.3906/mat-1707-15

D. M. Israfilov, A. Testici, Approximation in weighted generalized grand lebesgue space, Colloq. Math., 143, № 1, 113 – 126 (2016), https://doi.org/10.4064/cm6555-12-2015 DOI: https://doi.org/10.4064/cm6555-12-2015

D. M. Israfilov, A. Testici, Approximation in weighted generalized grand Smirnov classes, Stud. Sci. Math. Hung., 54, № 4, 471 – 488 (2017), https://doi.org/10.1556/012.2017.54.4.1378 DOI: https://doi.org/10.1556/012.2017.54.4.1378

D. Israfilov, V. Kokilashvili, S. Samko, Approximation in weighted Lebesgue and Smirnov spaces with variable exponents, Proc. A. Razmadze Math. Inst., 143, 25 – 35 (2007).

G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monogr., AMS (1969). DOI: https://doi.org/10.1090/mmono/026

I. Ibragimov, D. I. Mamedkhanov, A constructive characterization of a certain class of functions, Soviet Math. Dokl., 4, 820 – 823 (1976).

J. E. Andersson, On the degree of polynomial approximation in $E^p(D)$, J. Approx. Theory, 19, 61 – 68 (1977), https://doi.org/10.1016/0021-9045(77)90029-6 DOI: https://doi.org/10.1016/0021-9045(77)90029-6

J. L.Walsh, H. G. Russel, Integrated continuity conditions and degree of approximation by polynomials or by bounded analytic functions, Trans. Amer. Math. Soc., 92, 355 – 370 (1959), https://doi.org/10.2307/1993161 DOI: https://doi.org/10.1090/S0002-9947-1959-0108595-3

L. Greco, A remark on equality $mathrm{det} Df = mathrm{Det} Df $, Different. and Integral Equat., 6, 1089 – 1100 (1993).

L. Greco, T. Iwaniec, C. Sbordone, Inverting the $p$-harmonic operator, Manuscripta Math., 92, 249 – 258 (1997), https://doi.org/10.1007/BF02678192

M. Carozza, C. Sbordone, The distance to $L^{infty}$ in some function spaces and applications, Different. and Integral Equat., 10, 599 – 607 (1997), https://doi.org/10.1007/BF02678192 DOI: https://doi.org/10.1007/BF02678192

P. K. Suetin, Series of Faber Polynomials, Gordon and Breach Sci. Publ., New York (1998).

P. L. Duren, Theory of $H^p$ spaces, Acad. Press (1970).

R. Akgun, Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth, Georg. Math. J., 18, 203 – 235 (2011). DOI: https://doi.org/10.1515/gmj.2011.0022

R. Akgun, D. M. Israfilov, Approximation and moduli of fractional orders in Smirnov Orlicz classes, Glas. Math., 43, 121 – 136 (2008), https://doi.org/10.3336/gm.43.1.09 DOI: https://doi.org/10.3336/gm.43.1.09

S. Y. Alper, Approximation in the mean of analytic functions of class $E^p$ , (in Russian), Investigations on the Modern Problems of the Function Theory of a Complex Variable, Gos. Izdat. Fiz.-Mat. Lit., Moscow (1960), p. 272 – 286.

S. Z. Jafarov, Approximation by rational functions in Smirnov – Orlicz classes, J. Math. Anal. and Appl., 379, 870 – 877 (2011), https://doi.org/10.1016/j.jmaa.2011.02.027 DOI: https://doi.org/10.1016/j.jmaa.2011.02.027

S. Z. Jafarov, On approximation of functions by p-Faber – Laurent rational functions, Complex Variable and Elliptic Equations, 60, № 3, 416 – 428 (2015), https://doi.org/10.1080/17476933.2014.940928 DOI: https://doi.org/10.1080/17476933.2014.940928

S. Z. Jafarov, Approximation by polynomials and rational functions in Orlicz classes, J. Comput. Anal. and Appl., 13, 953 – 962 (2011).

S. Z. Jafarov, The inverse theorem of approximation theory in Smirnov – Orlicz spaces, Math. Inequal. Appl., 15, № 4, 835 – 844 (2012), https://doi.org/10.7153/mia-15-71 DOI: https://doi.org/10.7153/mia-15-71

T. Iwaniec, C. Sbordone, On integrability of the Jacobian under minimal hypotheses, Arch. Ration. Mech. and Anal., 119, 129 – 143 (1992), https://doi.org/10.1007/BF00375119 DOI: https://doi.org/10.1007/BF00375119

V. Kokilashvili, Boundedness criteria for singular integrals in weighted grand Lebesgue spaces, J. Math. Sci., 170, № 1, 20 – 33 (2010), https://doi.org/10.1007/s10958-010-0076-x DOI: https://doi.org/10.1007/s10958-010-0076-x

V. M. Kokilashvili, On analytic functions of Smirnov – Orlicz classes, Stud. Math., 31, 43 – 59 (1968), https://doi.org/10.4064/sm-31-1-43-59 DOI: https://doi.org/10.4064/sm-31-1-43-59

Published
20.07.2021
How to Cite
TesticiA. “Approximation by Rational Functions on Doubly Connected Domains in Weighted Generalized Grand Smirnov Classes”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 7, July 2021, pp. 964 -78, doi:10.37863/umzh.v73i7.559.
Section
Research articles