Approximation of Periodic Functions Belonging to $W(L^{r},\xi (t),(\beta \ge 0))$-Class By $(C^{1}\cdot T)$ Means of Fourier Series

Abstract

Various investigators such as Khan [3], Qureshi [8–10], Qureshi and Nema [11], Leindler [6] and Chandra [1] have determined the degree of approximation of functions belonging to the classes $W(L^{r},\xi (t)), Lip (\xi (t), r), Lip (\alpha , r)$ and $Lip \alpha$ using different summability methods with monotonocity conditions. Recently, Lal [5] has determined the degree of approximation of the functions belonging to $Lip \alpha$ and $W(L^{r},\xi (t))$ classes by using Cesàro-Nörlund $(C^{1}\cdot N_{p})$—summability with non-increasing weights $\{p_{n}\}$. In this paper, we shall determine the degree of approximation of 2$\pi$-periodic function f belonging to the function classes $Lip\alpha$ and $W(L^{r},\xi (t))$ by $C^{1}\cdot T$—means of Fourier series of f. Our theorems generalize the results of Lal [5], and we also improve these results in the light of [7, 12, 13]. From our results, we derive some corollaries also.