Trigonometric Approximation of Signals (Functions) Belonging to the <i>Lip</i>(<i>ξ</i>(<i>t</i>),<i>r</i>),(r＞1)-Class by (<i>E</i>,<i>q</i>) (<i>q</i>＞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 Lip α , Lip ( α , r ) , Lip ( ξ ( t ), r ) and W ( L r , 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 [1 7 ] 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 ].