Trigonometric Approximation of Signals (Functions) belonging to the W(Lr, $(t)), (r>=1)- class by (E,q) (q>0) -means of the conjugate series of its Fourier series

Abstract

Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. Various investigators such as Khan ([1]-[5]), Chandra [6, 7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [12], Mittal, Rhoades and Mishra [13], Mittal and Mishra [14], Rhoades et al. [15] have determined the degree of approximation of 2п- periodic signals (functions) belonging to various classes Lip a, Lip(a,r), Lip($(t),r), and W(Lr, $(t)), (r>=1) of functions through trigonometric Fourier approximation (TFA) using different summability matrices with monotone rows. Recently, Mittal et al. [16], Mishra [10], Mishra and Mishra [17] have obtained the degree of approximation of signals belonging to Lip(a,r)-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 W(Lr, $(t)), (r>=1)-  class by  summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Chandra [7], Shukla [21] and Mishra et al. [11].