Approximation of conjugate of functions belonging to weighted Lipschitz class W(Lp,ξ(t)) by Hausdorff means of conjugate Fourier series

摘要

In this paper, we review the previous work done on the degree of approximation of f˜, conjugate of a 2π-periodic function f belonging to the weighted W(Lp,ξ(t))-class and its subclasses such as Lip(ξ(t),p),Lip(α,p)andLipα, and determine the degree of approximation of f˜ by using Hausdorff means of conjugate Fourier series of f. Since (C,1), the Cesàro matrix of order 1, and (E,q), the Euler matrix of order q>0, are Hausdorff matrices, and the product of two Hausdorff matrices is also a Hausdorff matrix [B.E. Rhoades, N.K. Sharma, Spectral results for some Hausdorff matrices, Acta Sci. Math. 44 (1982) 359–364, p. 360, Theorem 1], our theorems generalize and improve some of the previous results. Some corollaries have also been deduced from our results.