ON APPROXIMATION BY NÖRLUND AND RIESZ SUBMETHODS IN VARIABLE EXPONENT LEBESGUE SPACES

Year 2018, Volume: 67 Issue: 1, 46 - 59, 01.02.2018

Uğur Değer

https://doi.org/10.1501/Commua1_0000000829

Cited By: 1

Abstract

Abstract. In this study the results on the degree of approximation by the Nörlund and the Riesz submethods of the partial sums of their Fourier series of functions where in the variable exponent Lebesgue spaces are given by weakening the monotonicity conditions of sequences in the submethods. Therefore the results given in G¸ven and Israfilov (2010) are generalized according to · both the monotonicity conditions and both the methods.

Keywords

Trigonometric approximation, generalized Lebesgue space, Nörlundsubmethod

References

• Quade, E. S., Trigonometric approximation in the mean, Duke Mathematical Journal. 3 (1937), 529–542.
• Sahney, B. N. and Rao, V. V. G., Error bounds in the approximation of functions, Bull. Austral. Math. Soc. 6 (1972), 11–18.
• Mohapatra, R. N. and Russell, D. C., Some direct and inverse theorems in approximation of functions, J. Austral. Math. Soc.(Ser. A). 34 (1983), 143–154.
• Chandra, P., Approximation by Nörlund operators, Mat. Vestnik. 38 (1986), 263–269.
• Chandra, P., Functions of classes Lpand Lip( ; p) and their Riesz means, Riv. Mat. Univ. Parma. (4)12 (1986), 275–282.
• Armitage, D. H. and Maddox, I. J., A new type of Cesáro mean, Analysis. 9 (1989), 195–204.
• Mazhar, S. M. and Totik, V., Approximation of continuous functions by T - means of Fourier series, J. Approx. Theory. (60)2 (1990), 174–182.
• Chandra, P., A note on degree of approximation by Nörlund and Riesz operators, Mat. Vestnik. 42 (1990), 9–10.
• Kováµcik, O. and Rákosnik, J., On spaces L and Wk;p(x), Czech. Math. J. 41(116) no.4 (1991), 592–618.
• Ky, N. X., Moduli of mean smoothness and approximation with Ap-weights, Annales Univ. Sci. Budapest. 40 (1997), 37–48.
• Edmunds, D. E., Lang, J. and Nekvinda, A., On Lp(x)norms, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 1981(455) (1999), 219— 225.
• Osikiewicz, J. A., Equivalance results for Cesáro submethods, Analysis. 20 (2000), 35–43.
• Leindler, L., On the uniform convergence and boundedness of a certain class of sine series, Analysis Mathematica. 27 (2001), 279–285.
• Fan, X. and Zhao,D., On the spaces L ( )and Wm;p(x)( ), J. Math. Anal. Appl. 263 (2001), 424–446.
• Chandra, P., Trigonometric approximation of functions in Lp-norm, Journal Of Mathematical Analysis and Applications. 275) (2002), 13–26.
• Diening, L. and Ruµziµcka, M., Calderón-Zygmund operators on generalized Lebesgue spaces Lp( )and problems related to *uid dynamics, J. reine angew. Math. 563 (2003), 197–220.
• Leindler, L., Trigonometric approximation in Lp-norm, Journal of Mathematical Analysis and Applications, 302 (2005), 129–136.
• Güven, A. and Isra…lov, D. M., Trigonometric approximation in generalized Lebesgue spaces Lp(x), J. Math. Inequal. 4(2) (2010), 285–299.
• Deµger, U., Daµgadur, ·I. and Küçükaslan, M., Approximation by trigonometric polynomials to functions in Lpnorm, Proc. Jangjeon Math. Soc. (15)2 (2012), 203–213.
• Mohapatra, R. N. and Szal, B., On Trigonometric approximation of functions in the Lp-norm, arXiv:1205.5869v1 [math.CA] (2012).
• Deµger, U. and Kaya, M., On the approximation by Cesáro submethod, Palestine Journal of Mathematics. (4)1 (2015), 44–56.
• Current address : U¼gur De˜ger: Mersin University, Faculty of Science and Literature, Depart- ment of Mathematics, 33343 Mersin - TURKEY.