On the Approximation of Generalized Lipschitz Function by Euler Means of Conjugate Series of Fourier Series

Approximation theory is a very important field which has various applications in pure and applied mathematics. The present study deals with a new theorem on the approximation of functions of Lipschitz class by using Euler's mean of conjugate series of Fourier series. In this paper, the degree of approximation by using Euler's means of conjugate of functions belonging to Lip (ξ(t), p) class has been obtained. Lipα and Lip (α, p) classes are the particular cases of Lip (ξ(t), p) class. The main result of this paper generalizes some well-known results in this direction.


Introduction and Definitions
Let be periodic with period 2 and integrable in the sense of Lebesgue. The Fourier series associated with at the point is given by with partial sums ( ; ). The conjugate series of (1) is given by with partial sums̃( ; ). Throughout this paper, we call (2) as conjugate series of Fourier series of function . If is Lebesgue integrable, theñ exists for almost all (Hardy [1], page 131).̃( ) is called the conjugate function of ( ).
∞ -norm of a function : → is defined by 2 The Scientific World Journal -norm is defined by The degree of approximation of a function : → by a trigonometric polynomial of order under sup norm ‖ ‖ ∞ is defined by ( [1], page 114-115]) and ( ) of a function ∈ is given by Let { } be the sequence of partial sums of the given series ∑ ∞

=0
. Then, for > 0, the Euler ( , ) means of { } are defined to be The series is said to be Euler ( , ) summable to provided that the sequence { } converges to as → +∞.
We write
Alexits [8] proved the following theorem concerning the degree of approximation of a function ∈ Lip by the ( , ) means of its Fourier series.
Theorem A. If a periodic function ∈ Lip , 0 < ≤ 1, then the degree of approximation of the ( , ) means of its Fourier series for 0 < < ≤ 1 is given by and for 0 < ≤ ≤ 1 is given by where ( ) are the ( , ) means of the partial sums of (2).
Later on, Hölland et al. [9] extended Theorem A to functions belonging to * [0, 2 ], the class of 2 -periodic continuous functions on [0, 2 ], using Nörlund means of Fourier series. Their theorem is as follows.
Theorem B. If ( ) is the modulus of continuity of ∈ * [0, 2 ], then the degree of approximation of by the Nörlund means of the Fourier series for f is given by where are the ( , ) means of Fourier series of .
Hölland et al. [9] have shown that Theorem B reduces to Theorem A if we deal with Cesàro means of order and consider a function ∈ Lip , 0 < ≤ 1. Working in same direction we prove the following theorem.

Theorem 1. If : → is a 2 periodic, Lebesgue integrable and belonging to Lip ( ( ), ) for > 1 and if
conditions (16) of the conjugate series (2) is given bỹ In order to prove our theorem, we need the following lemma.

Corollaries
The following corollaries may be derived from our theorem.