Abstract
Degree of approximation of functions of different classes has been studied by several researchers by different summability methods. In the proposed paper, we have established a new theorem for the approximation of a signal (function) belonging to the W(Lr,ξ(t))-class by (N¯,pn,qn)(E,s)-product summability means of a Fourier series. The result obtained here, generalizes several known theorems.
Highlights
The theory of summability arose from the process of summation of series and the significance of the concept of summability has been rightly demonstrated in varying contexts, e.g. in Fourier analysis, approximation theory and fixed point theory and many other fields
By generalized Hö lders inequality and Minkowski’s inequality, Mishra, Sonavane, and Mishra (2013) have proved Lr approximation of signals belonging to W(Lr, (t))-class by C1.Np-summability means of conjugate series of Fourier series
Mishra and Sonavane (2015) has proved approximation of functions belonging to the Lipschitz class by product mean (N, pn)(E, 1) of Fourier series
Summary
The theory of summability arose from the process of summation of series and the significance of the concept of summability has been rightly demonstrated in varying contexts, e.g. in Fourier analysis, approximation theory and fixed point theory and many other fields. The error approximation of periodic functions belonging to various Lipschitz classes through summability method is an active area of research in the last decades. By generalized Hö lders inequality and Minkowski’s inequality, Mishra, Sonavane, and Mishra (2013) have proved Lr approximation of signals belonging to W(Lr, (t))-class by C1.Np-summability means of conjugate series of Fourier series.
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