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 class has been obtained. and classes are the particular cases of class. The main result of this paper generalizes some well-known results in this direction.

1. Introduction and Definitions

Letbe periodic with periodand integrable in the sense of Lebesgue. The Fourier series associated withat the pointis given by with partial sums. The conjugate series of (1) is given by with partial sums. Throughout this paper, we call (2) as conjugate series of Fourier series of function. Ifis Lebesgue integrable, then exists for almost all(Hardy [1], page 131).is called the conjugate function of.

A functionif

,consider that if

(Definitionof Chandra [2]).

Given a positive increasing function,, whereis a positive number independent ofand.

In case, thencoincides with. Ifin, then it coincides with.

-norm of a functionis defined by

-norm is defined by

The degree of approximation of a functionby a trigonometric polynomialof orderunder sup norm is defined by ([1], page 114-115]) andof a functionis given by

Letbe the sequence of partial sums of the given series. Then, for, the Euler () means ofare defined to be

The series is said to be Euler () summable toprovided that the sequenceconverges toas.

We write

2. Main Theorem

Hardy [1] established a theorem on (), () summability of the series. Harmonic summability is weaker than () summability. Iyengar [3] proved a theorem on harmonic summability of a Fourier series. The result of Iyengar [3] has been generalized by several researchers like Siddiqi [4], Pati [5], Lal and Kushwaha [6], and Rajagopal [7], for Nörlund means.

Alexits [8] proved the following theorem concerning the degree of approximation of a functionby the () means of its Fourier series.

Theorem A. If a periodic function,, then the degree of approximation of the () means of its Fourier series foris given by and foris given by whereare the () means of the partial sums of (2).

Later on, Hölland et al. [9] extended Theorem A to functions belonging to, the class of-periodic continuous functions on [], using Nörlund means of Fourier series. Their theorem is as follows.

Theorem B. Ifis the modulus of continuity of, then the degree of approximation ofby the Nörlund means of the Fourier series for f is given by whereare themeans of Fourier series of.

Hölland et al. [9] have shown that Theorem B reduces to Theorem A if we deal with Cesàro means of orderand consider a function,. Working in same direction we prove the following theorem.

Theorem 1. Ifis aperiodic, Lebesgue integrable and belonging toforand if conditions (16) and (17) hold uniformly in, then degree of approximation of, conjugate of, by Euler () mean of the conjugate series (2) is given by

In order to prove our theorem, we need the following lemma.

Lemma 2. If, then

Proof. We have sincewhen. Therefore,

Proof of Theorem 1. Thepartial sum of the conjugate series of the Fourier series (2) is given by Taking Euler () means, we get using Lemma 2.
Clearly, Hence, by Minkowski’s inequality, Then,.
Using Hölder’s inequality,, condition (16),, lemma, and second mean value theorem for integrals, we have where Now, Using Hölder’s inequality,, and condition (17), we have Combining (24) with (31), we have which completes the proof of the theorem.

3. Corollaries

The following corollaries may be derived from our theorem.

Corollary 3. If, then the degree of approximation of a function, conjugate of,, by Euler’s meansof the conjugate series of the Fourier series (2) is given by

Corollary 4. Ifin Corollary 3, then, for,