APPROXIMATING CLASSES OF FUNCTIONS DEFINED BY A GENERALISED MODULUS OF SMOOTHNESS

Year 2013, Volume: 1 Issue: 2, 84 - 89, 01.12.2013

Faton M. Berısha Nimete Sh.berısha

Abstract

In the present paper, we use a generalised shift operator in order to define a generalised modulus of smoothness. By its means, we definegeneralised Lipschitz classes of functions, and we give their constructive characteristics. Specifically, we prove certain direct and inverse types theorems inapproximation theory for best approximation by algebraic polynomials

Keywords

generalised Lipschitz classes, generalised modulus of smoothness, asymmetric generalised shift operator, theorem of coincidence of classes

References

• F. M. Berisha. On coincidence of classes of functions defined by a generalized modulus of smoothness and the appropriate inverse theorem. Math. Montisnigri, 9:15–36, 1998.
• N. Sh. Berisha and F. M. Berisha. Approximating classes of functions defined by operators of differentiation or operators of generalised translation by means of algebraic polynomials. Int. J. Math. Anal. (Ruse), 6(55):2709–2729, 2012.
• M. K. Potapov and F. M. Berisha. Approximation of classes of functions defined by a gener- alized k-th modulus of smoothness. East J. Approx., 4(2):217–241, 1998.
• M. K. Potapov and F. M. Berisha. Direct and inverse theorems of approximation theory for a generalized modulus of smoothness. Anal. Math., 25(3):187–203, 1999.
• M. K. Potapov and F. M. Berisha. O svyazi mezhdu r-im obobshchennym medulem gladkosti i nailuchshimi priblizheniyami algebraicheskimi mnogochlenami. Fundam. Prikl. Mat., 5(2):563– 587, 1999. F. M. Berisha, Faculty of Mathematics and Sciences, University of Prishtina, N¨ena
• Terez¨e 5, 10000 Prishtin¨e, Kosovo E-mail address: [e-mail] N. Sh. Berisha, Faculty of Mathematics and Sciences, University of Prishtina, N¨ena
• Terez¨e 5, 10000 Prishtin¨e, Kosovo E-mail address: [e-mail]