Trigonometric Approximation of Signals (Functions) Belonging toW(Lr, ξ(t))Class by Matrix

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,r)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,r)- 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 αandW(Lr,ξ(t)) classes by using Cesáro-Nörlund(C1·Np)summability without monotonicity condition on{pn}, 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).

On approximation to functions in the $W(L^{p}, \xi(t))$ class by a new matrix mean

“…≡ (a m,k ) is a Nörlund matrix, then the (C 1 · T ) means give us the Cesàro-Nörlund (C 1 · N p ) means. Accordingly, our main theorems coincide with Theorem 2.3 and Theorem 2.4 in [23]. Moreover, our main results generalize the main results in [8] and [23].…”

“…Accordingly, our main theorems coincide with Theorem 2.3 and Theorem 2.4 in [23]. Moreover, our main results generalize the main results in [8] and [23]. , then the (C 1 · T ) means give us the (C, 1)(E, 1) product means.…”

“…Nigam has studied the same problem for the (C, 1)(E, q) means, which are much more general than the (C, 1)(E, 1) means in [16]. Sing, Mittal and Sonker have generalized the results of Lal [8] in [23]. Therefore, taking into account this generalization of the function classes, we shall give two theorems on degree of approximation to functions belonging to the classes W (L p , ξ(t)) and Lipα by the (C 1 · T ) matrix means, being more general than (C, 1)(E, 1), (C 1 · N p ) and (C, 1)(E, q) means given in [6], [8,23] and [16], respectively.…”