Trigonometric Approximation of Signals (Functions) Belonging to the Lip(ξ(t),r),(r>1)-Class by (E,q) (q>0)-Means of the Conjugate Series of Its Fourier Series

Abstract

Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades et al. [14] have determined the degree of approximation of 2π-periodic signals (functions) belonging to various classes Lipα, Lip(α,r), Lip(ξ(t),r) and W(Lr,ζ(t)) of functions through trigonometric Fourier approximation (TFA) using different summability matrices with monotone rows. Recently, Mittal et al. [15], Mishra and Mishra [16], Mishra [17] have obtained the degree of approximation of signals belonging to -class by general summability matrix, which generalizes the results of Leindler [8] and some of the results of Chandra [7] by dropping monotonicity on the elements of the matrix rows (that is, weakening the conditions on the filter, we improve the quality of digital filter). In this paper, a theorem concerning the degree of approximation of the conjugate of a signal (function) f belonging to Lip(ξ(t),r) class by (E,q) summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Chandra [7] and Shukla [18].

Share and Cite:

V. Mishra, H. Khan, I. Khan, K. Khatri and L. Mishra, "Trigonometric Approximation of Signals (Functions) Belonging to the Lip(ξ(t),r),(r>1)-Class by (E,q) (q>0)-Means of the Conjugate Series of Its Fourier Series," Advances in Pure Mathematics, Vol. 3 No. 3, 2013, pp. 353-358. doi: 10.4236/apm.2013.33050.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] H. H. Khan, “On Degree of Approximation to a Functions Belonging to the Class Lip(α,p),” Indian Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 1974, pp. 132-136.
[2] H. H. Khan, “On the Degree of Approximation to a Function by Triangular Matrix of Its Fourier Series I,” Indian Journal of Pure and Applied Mathematics, Vol. 6, No. 8, 1975, pp. 849-855.
[3] H. H. Khan, “On the Degree of Approximation to a Function by Triangular Matrix of Its Conjugate Fourier Series II,” Indian Journal of Pure and Applied Mathematics, Vol. 6, No. 12, 1975, pp. 1473-1478.
[4] H. H. Khan, “A Note on a Theorem Izumi,” Communications De La Faculté Des Sciences Mathématiques Ankara (TURKEY), Vol. 31, 1982, pp. 123-127.
[5] H. H. Khan and G. Ram, “On the Degree of Approximation,” Facta Universitatis Series Mathematics and Informatics (TURKEY), Vol. 18, 2003, pp. 47-57.
[6] P. Chandra, “A Note on the Degree of Approximation of Continuous Functions,” Acta Mathematica Hungarica, Vol. 62, No. 1-2, 1993, pp. 21-23.
[7] P. Chandra, “Trigonometric Approximation of Functions in -Norm,” Journal of Mathematical Analysis and Applications, Vol. 275, No. 1, 2002, pp. 13-26. doi:10.1016/S0022-247X(02)00211-1
[8] L. Leindler, “Trigonometric Approximation in Lp-Norm,” Journal of Mathematical Analysis and Applications, Vol. 302, No. 1, 2005, pp. 129-136. doi:10.1016/j.jmaa.2004.07.049
[9] V. N. Mishra, H. H. Khan and K. Khatri, “Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator,” Applied Mathematics, Vol. 2, No. 12, 2011, pp. 1448-1452. doi:10.4236/am.2011.212206
[10] V. N. Mishra, “On the Degree of Approximation of Signals (Functions) Belonging to the Weighted W(Lp,ξ(t)),(p≥1) -Class by Almost Matrix Summability Method of Its Conjugate Fourier Series,” International Journal of Applied Mathematics and Mechanics, Vol. 5, No. 7, 2009, pp. 16-27.
[11] M. L. Mittal, U. Singh, V. N. Mishra, S. Priti and S. S. Mittal, “Approximation of functions belonging to Lip(ξ(t),r),(r>1)-Class by means of conjugate Fourier series using linear operators,” Indian Journal of Mathematics, Vol. 47, No. 2-3, 2005, pp. 217-229.
[12] M. L. Mittal, B. E. Rhoades and V. N. Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(Lp,ξ(t)),(p≥1) -Class by linear operators,” International Journal of Mathematics and Mathematical Sciences, Vol. 2006, 2006, Article ID: 53538. doi:10.1155/IJMMS/2006/53538
[13] M. L. Mittal and V. N. Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(Lp,ξ(t)),(p≥1) -Class by Almost Matrix Summability Method of Its Fourier Series,” International Journal of Mathematical Sciences and Engineering Applications, Vol. 2, No. 4, 2008, pp. 285-294.
[14] B. E. Rhoades, K. Ozkoklu and I. Albayrak, “On Degree of Approximation to a Functions Belonging to the Class Lipschitz Class by Hausdroff Means of Its Fourier Series,” Applied Mathematics and Computation, Vol. 217, No. 16, 2011, pp. 6868-6871. doi:10.1016/j.amc.2011.01.034
[15] M. L. Mittal, B. E. Rhoades, V. N. Mishra and U. Singh, “Using Infinite Matrices to Approximate Functions of Class Lip(α,p) Using Trigonometric Polynomials,” Journal of Mathematical Analysis and Applications, Vol. 326, No. 1, 2007, pp. 667-676. doi:10.1016/j.jmaa.2006.03.053
[16] V. N. Mishra and L. N. Mishra, “Trigonometric Approximation of Signals (Functions) in Lp( p≥1)-Norm,” International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 19, 2012, pp. 909-918.
[17] V. N. Mishra, “Some Problems on Approximations of Functions in Banach Spaces,” Ph.D. Thesis, Indian Institute of Technology, Roorkee, 2007.
[18] R. K. Shukla, “Certain Investigations in the theory of Summability and that of Approximation,” Ph.D. Thesis, V.B.S. Purvanchal University, Jaunpur, 2010.
[19] G. H. Hardy, “Divergent Series,” Oxford University Press, Oxford, 1949.
[20] A. Zygmund, “Trigonometric Series, Vol. I,” 2nd Edition, Cambridge University Press, Cambridge, 1959.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.