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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

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Author(s): Vishnu Narayan Mishra | Huzoor H. Khan | Idrees A. Khan | Kejal Khatri | Lakshmi N. Mishra

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 03;
Issue: 03;
Start page: 353;
Date: 2013;
Original page

Keywords: Signals; Conjugate Fourier Series; Trigonometric Fourier Approximation; Degree of Approximation; Lip(ξ(t) | r)-Class; (E | q) Summability

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].
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