International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 53538, 10 pages
doi:10.1155/IJMMS/2006/53538

Approximation of signals (functions) belonging to the weighted W(Lp,ξ(t))-class by linear operators

M. L. Mittal,1 B. E. Rhoades,2 and Vishnu Narayan Mishra3

1Department of Mathematics, IIT Roorkee, Roorkee 247667, Uttaranchal, India
2Department of Mathematics, Indiana University, Bloomington 47405, IN, USA
3Department of Mathematics, IIT Roorkee, Roorkee 247667, Uttaranchal, India

Received 23 May 2006; Accepted 2 October 2006

In memory of Professor Brian Kuttner (1908–1992)

Copyright © 2006 M. L. Mittal 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

Mittal and Rhoades (1999–2001) and Mittal et al. (2006) have initiated the studies of error estimates En(f) through trigonometric Fourier approximations (TFA) for the situations in which the summability matrix T does not have monotone rows. In this paper, we determine the degree of approximation of a function f˜, conjugate to a periodic function f belonging to the weighted W(Lp,ξ(t))-class (p≥1), where ξ(t) is nonnegative and increasing function of t by matrix operators T (without monotone rows) on a conjugate series of Fourier series associated with f. Our theorem extends a recent result of Mittal et al. (2005) and a theorem of Lal and Nigam (2001) on general matrix summability. Our theorem also generalizes the results of Mittal, Singh, and Mishra (2005) and Qureshi (1981-1982) for Nörlund (Np)-matrices.