Euler-Hausdorff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class

Shyam Lal,Abhishek Mishra

doi:10.1186/1029-242x-2013-59

Shyam Lal, Abhishek Mishra

Open Access

https://doi.org/10.1186/1029-242x-2013-59

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Abstract

In this paper, three new estimates for the degree of approximation of a function ˜ f ,t he conjugate of a function f belonging to classes Lip α and Lip(ξ , r), r ≥ 1, by E (q) · � H summability operator of conjugate series of the Fourier series have been determined. MSC: 42A24; 41A25; 42B05; 42B08

Highlights

1 Introduction The degree of approximation of the function f ∈ Lip α and Lip(α, r), r ≥, classes have been determined by several investigators like Alexits [ ], Sahney and Goel [ ], Chandra [ ], Qureshi [, ] and Qureshi and Nema [ ]
After quite a good amount of work on the degree of approximation of functions by different summability means of its Fourier series, Lal and Singh [ ] established the degree of approximation of conjugates of Lip(α, r) functions by (C, )(E, ) product means of conjugate series of a Fourier Series in the following form
If f : R → R is a π -periodic and Lip(α, r) function, the degree of approximation of its conjugate function fby the (C, )(E, ) product means of conjugate series of the Fourier series of f satisfies

Summary

Introduction

The degree of approximation of the function f ∈ Lip α and Lip(α, r), r ≥ , classes have been determined by several investigators like Alexits [ ], Sahney and Goel [ ], Chandra [ ], Qureshi [ , ] and Qureshi and Nema [ ]. Nigam and Sharma have obtained the degree of approximation by the Karmata summability method [ ] and by (C, )(E, q) means of its Fourier series [ ] as follows. If a function f is π -periodic, Lebesgue integrable on [ , π] and belongs to Lip(ξ (t), r) class, its degree of approximation by (C, )(E, q) summability means of its Fourier series is given by

Results

Conclusion