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We prove two Theorems on approximation of functions belonging to Lipschitz class Lip(α,p) in Lp-norm using Cesàro submethod. Further we discuss few corollaries of our Theorems and compare them with the existing results. We also note that our results give sharper estimates than the estimates in some of the known results.

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimates En(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper, the first author continues the work in the direction for T to be a Np-matrix. We extend two theorems on summability matrix Np of Deger et al. (2012) where they have extended two theorems of Chandra (2002) using Cλ-method obtained by deleting a set of rows from Cesàro matrix C1. Our theorems also generalize two theorems of Leindler (2005) to Np-matrix which in turn generalize the result of Chandra (2002) and Quade (1937).

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 α and W(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).

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.