Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements

In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence and summability methods by regular summability matrices.

Trigonometric Approximation of Signals (Functions) Belonging toW(Lr, ξ(t))Class by Matrix(C1·Np)Operator

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).