Glasnik Matematicki, Vol. 47, No. 2 (2012), 401-413.

APPROXIMATION OF PERIODIC FUNCTIONS IN WEIGHTED ORLICZ SPACES

Yunus E. Yildirir

Abstract.   In the present work we prove some direct theorems of the approximation theory in the weighted Orlicz spaces with weights satisfying so called Muckenhoupt's condition and we obtain some estimates for the deviation of a function in the weighted Orlicz spaces from the linear operators constructed on the basis of its Fourier series.

2010 Mathematics Subject Classification.   41A10, 42A10.

Key words and phrases.   Direct theorem, weighted Orlicz space, Muckenhoupt weight, modulus of smoothness.

DOI: 10.3336/gm.47.2.13

References:

1. R. Akgun and D. M. Israfilov, Approximation in weighted Orlicz spaces, Math. Slovaca 61 (2011), 601-618.
   MathSciNet     CrossRef

2. R. Akgun and D. M. Israfilov, Approximation and moduli of fractional order in Smirnov-Orlicz classes, Glas. Mat. Ser. III 43(63) (2008), 121-136.
   MathSciNet     CrossRef

3. R. Akgun, Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst. 152 (2010), 1-18.
   MathSciNet    

4. R. A. De Vore and G. G. Lorentz, Constructive approximation, Springer-Verlag, Berlin, 1993.
   MathSciNet    

5. Z. Ditzian and V. Totik, Moduli of smoothness, Springer-Verlag, New York, 1987.
   MathSciNet     CrossRef

6. I. Genebashvili, A. Gogatishvili, V. Kokilashvili and M. Krbec, Weight theory for integral transforms on spaces of homogeneous type, Longman, Harlow, 1998.
   MathSciNet    

7. E. A. Haciyeva, Investigation of the properties of functions with quasimonotone Fourier coefficients in generalized Nikolskii-Besov spaces, Author's summary of dissertation, Tbilisi, 1986 (Russian).

8. D. M. Israfilov and A. Guven, Approximation by trigonometric polynomials in weighted Orlicz spaces, Studia Math. 174 (2006), 147-168.
   MathSciNet     CrossRef

9. D. M. Israfilov and R. Akgun, Approximation in weighted Smirnov-Orlicz classes, J. Math. Kyoto Univ. 46 (2006), 755-770.
   MathSciNet    

10. D. M. Israfilov, B. Oktay, and R. Akgun, Approximation in Smirnov- Orlicz classes, Glas. Mat. Ser. III 40(60) (2005), 87-102.
    MathSciNet     CrossRef

11. V. M. Kokilashvili, On analytic functions of Smirnov-Orlicz spaces, Studia Math. 31 (1968), 43-59.
    MathSciNet    

12. V. M. Kokilashvili, A direct theorem on mean approximation of analytic functions by polynomials, Dokl. Akad. Nauk SSSR 10 (1969), 411-414.

13. V. Kokilashvili, A note on extrapolation and modular inequalities, Proc. of A. Razmadze Math. Inst. 150 (2009), 91-97
    MathSciNet    

14. M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex functions and Orlicz spaces, Noordhoff Ltd., Groningen, 1961.
    MathSciNet    

15. N. X. Ky, On approximation by trigonometric polynomials in L_u^p-spaces, Studia Sci. Math. Hungar. 28 (1993), 183-188.
    MathSciNet    

16. H. N. Mhaskar, Introduction to the theory weighted polynomial approximation, World Sci., River Edge, 1996.
    MathSciNet    

17. V. G. Ponomarenko, Approximation of periodic functions in Orlicz space, Translated from Sibirskii Matematicheskii Zhurnal 7 (1966), 1337-1346.
    MathSciNet    

18. A. R-K. Ramazanov, On approximation by polynomials and rational functions in Orlicz spaces, Anal. Math. 10 (1984), 117-132.
    MathSciNet     CrossRef

19. M. M. Rao and Z. D. Ren, Application of Orlicz spaces, Dekker, 2002.
    MathSciNet     CrossRef

20. M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Marcel Dekker, New Tork, 1991.
    MathSciNet    

21. K. Runovski, On Jackson type inequality in Orlicz classes, Rev. Mat. Complut. 14 (2001), 395-404.
    MathSciNet    

22. A. F. Timan, Theory of approximation of functions of a real variable, Pergamon press and Macmillan, Oxford, 1963.
    MathSciNet    

23. Y. E. Yildirir and D. M. Israfilov, The properties of convolution type transforms in weighted Orlicz spaces, Glas. Mat. Ser. III 45(65) (2010), 461-474.
    MathSciNet     CrossRef

24. G. Wu, On approximation by polynomials in Orlicz spaces, Approx. Theory Appl. 7 (1991), 97-110.
    MathSciNet