Approximation by rational functions in Morrey-Smirnov classes

Mohammad Ali, Suliman Mahmoud, Ahmed kinj


In this article, we investigate the direct problem of approximation theory in Morrey-Smirnov classes of analytic functions, defined on a doubly-connected domain bounded by two sufficiently smooth curves.


Dini-smooth curve; modulus of smoothness; Morrey space; Morrey-Smirnov classes; Rational approximation.

Full Text:



Bilalov, B., Gasymov, T. & Guliyeva, A. (2016). On the solvability

of the Riemann boundary value problem in Morrey- Hardy classes.

Turkish Journal of Mathematics, 40(5):1085-1101.

Bilalov, B. & Najafov, T. (2013). On basicity of systems

of generalized Faber polynomials. Jaen Journal on


Bilalov, B. & Guliyeva, A. (2014). On basicity of exponential

systems in Morrey-type spaces. International Journal of

Mathematics, 25(6):1450054 (10 pages).

Devore, R.A. & Lorentz, G.G. (1993). Constructive approximation.

Springer. New York.

Gaier, D. (1987). Lectures on complex approximation. Birkhäser

Verlag. Boston- Stuttgart.

Goluzin, G. (1969). Geometric theory of functions of a complex

variable. Vol 26, Translations of Mathematical Monographs.

Providence. Rhode Island.

Israfilov, D., Oktay, B. & Akgün, R. (2005). Approximation in

Smirnov- Orlicz classes. Glasnik mathematički, 40(1): 87-102.

Israfilov, D. & Tozman, N. (2008). Approximation by

polynomials in Morrey- Smirnov classes. East Journal on


Israfilov, D. & Tozman, N. (2011). Approximation in Morrey-

Smirnov classes.Azerbaijan Journal of Mathematics,1(1):99-113.

Jafarov, S. (2015). On approximation of functions by p- Faber

Laurent rational functions. Complex Variables and Elliptic

Equations, 60(3):416-428.

Keldysh. M. (1945). Sur la représentation par des séries de

polynômes des fonctionsd’une variable complexe dans de

domainesfermés (Russian). MatematicsSbornik,16(58):249–258.

Koç, H. (2016). Simultaneous approximation by polynomials in

Orlicz spaces generated by quasiconvex Young function. Kuwait

Journal of Science, 43(4): 18-31.

Kokilashvili, V. (1968). On approximation of analytic functions

from Ep classes. Trudy Tbiliss. Mat. im Razmadze Akad. Nauk

Gruzin: 82-102.

Kokilashvili, V. & Meskhi, A. (2008). Boundedness of maximal

and singular operators in Morrey spaces with variable exponent.

Armenian Journal of Mathematics,1(1):18-28.

Laverentiev, M. (1936). Sur les fonctionsd’une variable complexe

représentables par des series de polynomes. Hermann, Paris.

Markushevich, A. (1968). Theory of analytic functions, vol. 2

Izdatelstvo Nauka, Moscow.

Runge, C. (1885). Zur Theorie der eindeutigenanalytischenfunktio

nen. Acta Mathematica 6(1):229-244.

Samko, N. (2009). Weighted Hardy and singular operators in

Morrey spaces. Journal of Mathematics Analysis and Applications,


Suetin, P. (1998). Series of Faber polynomials. Gordon and Breach

Science Publishers, Amsterdam.

Walsh, J. (1998). Interpolation and approximation by rational

functions in the complex domain, Vol. 20 American Mathematical

Society Colloquium publications, Providence, Rhode Island.

Warschawski, E. (1932). Uber das ranverhalten der Ableitung

der Abildunggsfunktion bei Konfermer Abbildung. (Germany)

Mathematische Zeitschrift, 35(1):321-456.