Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes
DOI:
https://doi.org/10.37863/umzh.v73i7.559Ключові слова:
Doubly connected domai, modulus of smoothnes, Faber-Laurent serie, generalized grand Smirnov clas, Carleson curveАнотація
УДК 517.5
Наближення рацiональними функцiями для двозв’язних областей у зважених узагальнених великих класах Смiрнова
Нехай $G ⊂ ℂ$— двозв’язна область обмежена двома спрямними кривими Карлесона. У цiй роботi за допомогою вищого модуля гладкостi вивчається апроксимацiйнi властивостi рацiональних $p - ε$ функцiй Фабера –Лорана у пiдкласах зважених узагальнених великих класiв Смiрнова $E^{p),θ}(G,ω)$ аналiтичних функцiй.
Посилання
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
