Degrees of the approximations by some special matrix means of conjugate Fourier series

Radosława Kranz 1 , Aleksandra Rzepka 2 ,  and Ewa Sylwestrzak-Maślanka 1
  • 1 University of Zielona Góra, Faculty of Mathematics, Computer Science and Econometrics, 65-516, Zielona Góra, Poland
  • 2 University of Zielona Góra, Faculty of Mathematics, Computr Science and Econometrics, 65-516, Zielona Góra, Poland

Abstract

In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.

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  • [1] Zygmund A., Trigonometric series, Cambridge, 2002

  • [2] Leindler L., On the degree of approximation of continuous functions, Acta Math. Hungar., 2004, 104 (1-2), 105-113

  • [3] Szal B., A note on the uniform convergence and boundedness a generalized class of sine series, Commentat. Math., 2008, 48(1), 85-94

  • [4] Qureshi K., On the degree of approximation of functions belonging to the Lipschitz class by means of a conjugate series, Indian J. Pure Appl. Math., 1981, 12(9), 1120-1123

  • [5] Lal S., Nigam H. K., Degree of approximation of conjugate of a function belonging to Lip(ξ(t) , p) class by matrix summability means of conjugate Fourier series, Int. J. Math. Math. Sci., 2001, 27(9), 555-563

  • [6] Dhakal B. P., Approximation of the conjugate of function belonging to Lip class by (E, 1)(C, 1) means of the conjugate series of it’s Fourier series, Int. J. Innov. Res. Sci. Eng. Technol., 2013, 2(3), 836-840

  • [7] Lal S., Singh G. K., Degree of approximation of conjugate of Lip class function by K-summability means of conjugate series of a Fourier series, Tamkang J. Math., 2003, 34(4), 387-394

  • [8] Mishra V. N., Khan H. H., Khatri K., Degree of approximation of conjugate of signals (functions) by lower triangular matrix operator, App. Math., 2011, 2, 1448-1452

  • [9] Mishra L. N., Mishra V. N., Sonavane V., Trigonometric approximation of functions belonging to Lipschitz class by matrix (C1 · Np) operator of conjugate series of Fourier series, Adv. Difference Equ., 2013, 2013:127, 1-10

  • [10] Nigam H. K., Sharma A., On approximation of conjugate of functions belonging to different classes by product means, Int. J. Pure Appl. Math., 2012, 76(2), 303-316

  • [11] Rhoades B. E., The degree of approximation of functions, and their conjugates, belonging to several general Lipschitz classes by Hausdorff matrix means of the Fourier series and conjugate series of a Fourier series, Tamkang J. Math., 2014, 45(4), 389-395

  • [12] Sonker S., Singh U., Degree of approximation of the conjugate of signals (functions) belonging to Lip( r)-class by (C, 1)(E, q) means of conjugate trigonometric Fourier series, J. Inequal. Appl , 2012, 2012:278, 1-7

  • [13] Qureshi K., On the degree of approximation of functions belonging to the class Lip( p) by means of a conjugate series, Indian J. Pure Appl. Math., 1982, 13(5), 560-563

  • [14] Łenski W., Szal B., Approximation of functions belonging to the class Lp by linear operators, Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24

  • [15] Sain S., Singh U., Degree of approximation of function belonging to Lip((t), )-class by linear operators based on Fourier series, Boll. Unione Mat. Ital., 2016, 9, 495-504

  • [16] Łenski W., Szal B., Approximation of functions from Lp by linear operators of conjugate Fourier series, Banach Center Publications, 2011, 92, 237-247

  • [17] Totik V., On the strong approximation by the means of Fourier series I, Anal. Math., 1980, 6, 57-85

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