International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 964101, 11 pages
http://dx.doi.org/10.1155/2012/964101
Research Article

Trigonometric Approximation of Signals (Functions) Belonging to π‘Š ( 𝐿 π‘Ÿ , πœ‰ ( 𝑑 ) ) Class by Matrix ( 𝐢 1 β‹… 𝑁 𝑝 ) Operator

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

Received 22 March 2012; Revised 24 April 2012; Accepted 3 May 2012

Academic Editor: Jewgeni Dshalalow

Copyright © 2012 Uaday Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2π-periodic signals (functions) belonging to Lip ( 𝛼 , π‘Ÿ ) class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip ( 𝛼 , π‘Ÿ ) - class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lip α and π‘Š ( 𝐿 π‘Ÿ , πœ‰ ( 𝑑 ) ) classes by using Cesáro-Nörlund ( 𝐢 1 β‹… 𝑁 𝑝 ) summability without monotonicity condition on { 𝑝 𝑛 } , which in turn generalizes the results of Lal (2009). We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011).