Approximation of signals (functions) belonging to the weighted <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>W</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>-class by linear operators

Mittal, M. L.; Rhoades, B. E.; Mishra, Vishnu Narayan

doi:https://doi.org/10.1155/IJMMS/2006/53538

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 053538 | https://doi.org/10.1155/IJMMS/2006/53538

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

M. L. Mittal,¹B. E. Rhoades,²and Vishnu Narayan Mishra¹

Received23 May 2006

Accepted02 Oct 2006

Published29 Nov 2006

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.

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Copyright © 2006 Hindawi Publishing Corporation. 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.

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