On approximation in generalized Zygmund class

Abstract

Here, we estimate the degree of approximation of a conjugate function g˜{\tilde g} and a derived conjugate function g˜′{\tilde g'} , of a 2π-periodic function g∈Zrλg \in Z_r^\lambda , r ≥ 1, using Hausdorff means of CFS (conjugate Fourier series) and CDFS (conjugate derived Fourier series) respectively. Our main theorems generalize four previously known results. Some important corollaries are also deduced from our main theorems. We also partially review the earlier work of the authors in respect of order of the Euler-Hausdorff product method.

Keywords

• generalized zygmund (zrλ, r ≥ 1) class of function
• error approximation
• cfs (conjugate fourier series)
• cdfs (conjugate derived fourier series)
• generalized minkowski’s inequality
• hausdorff means
• primary: 42a10, 41a10
• secondary: 42b05, 42b08