TITLE:
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
AUTHORS:
Vishnu Narayan Mishra, Huzoor H. Khan, Idrees A. Khan, Kejal Khatri, Lakshmi N. Mishra
KEYWORDS:
Signals; Conjugate Fourier Series; Trigonometric Fourier Approximation; Degree of Approximation; Lip(ξ(t), r)-Class; (E, q) Summability
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.3,
May
16,
2013
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].