On the trigonometric approximation of functions in a weighted Lipschitz class

Abstract

Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) obtained the degree of approximation of $$\tilde{f}$$ , conjugate of a 2π periodic function $$f$$ belonging to the weighted Lipschitz class $$W(L_{r} ,\beta ,\xi )(r \ge 1)$$ , through the Cesaro-Norlund means of the conjugate Fourier series of f, which is an extension of the results of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Singh et al. (International Journal of Mathematics and Mathematical Sciences 2012:1–12, 2012). Zhang (Applied Mathematics and Computation 247:1139–1140, 2014) has pointed out that the conclusions of the theorem of Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) hold only for constant functions, and thus the results are trivial. In this paper, we redefine the problems of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) for $$f \in W(L_{r} ,\beta ,\xi )(r \ge 1)$$ and its conjugate $$\tilde{f}$$ . We obtain the degree of approximation through more general summability means under weighted norm which in turn resolve the issues raised by Zhang (Applied Mathematics and Computation 247:1139–1140, 2014). We also derive some corollaries from our results.