Search

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.

Abstract In the present work, a best approximation of a function f in Besov space using the generalized Nörlund–Hausdorff (Npq△H) $(N_{pq}\triangle _{H})$ product means, for the two different cases 1

Abstract In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class Z p ( ω ) $Z_{p}^{(\omega)}$ ( p ≥ 1 $p \ge1$ ) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results.

In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class $Z_{p}^{(\omega)}$ ( $p \ge1$ ) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results. [ABSTRACT FROM AUTHOR]

Rhoades [13] has obtained the degree of approximation of functions belonging to the weighted Lipschitz class W(Lp; ε (t)) by Hausdorff means of their Fourier series, where ε(t) is an increasing function. The first result of Rhoades [13] generalizes the result of Lal [2]. In a very recent paper Rhoades et al: [14] have obtained the degree of approximation of functions belonging to the Lipα class by Hausdorff means of their Fourier series and generalized the result of Lal and Yadav [7]. The authors in [14] have made some important remarks, namely, increasing nature of ε(t) alone is not sufficient to prove the results of Lal [2], Lal and Singh [6], Qureshi [11] and Rhoades [13]; and the condition 1= sinβ (t) = O(1/tβ); 1/n ≤ t ≤ π used by all these authors is not valid since sin t → 0 as t → π. They have also suggested a modification in the definition of weighted (Lp, ε (t)) - class and leave an open question for determining a correct set of conditions to prove the results of Rhoades [13]. We note that the same types of errors can also be seen in the papers of Lal [3, 4], Nigam [8, 9] and Nigam and Sharma [10]. Being motivated by the remarks of Rhoades et al: [14], in this paper, we determine the degree of approximation of functions belonging to the weighted (Lp, ε (t)) class by Hausdorff means of their Fourier series and rectify the above errors by using proper set of conditions. We also deduce some important corollaries from our result. [ABSTRACT FROM AUTHOR]