International Journal of Mathematics and Mathematical Sciences

1. Introduction

Khan [1, 2] has studied the degree of approximation of a function belonging to and classes by Nörlund and generalized Nörlund means. Working in the same direction Rhoades [3], Mittal et al. [4], Mittal and Mishra [5], and Mishra [6, 7] have studied the degree of approximation of a function belonging to class by linear operators. Thereafter, Nigam [8] and Nigam and Sharma [9] discussed the degree of approximation of conjugate of a function belonging to class and by product summability means, respectively. Recently, Rhoades et al. [10] have determined very interesting result on the degree of approximation of a function belonging to class by Hausdorff means. Summability techniques were also applied on some engineering problems like Chen and Jeng [11] who implemented the Cesàro sum of order and , in order to accelerate the convergence rate to deal with the Gibbs phenomenon, for the dynamic response of a finite elastic body subjected to boundary traction. Chen et al. [12] applied regularization with Cesàro sum technique for the derivative of the double layer potential. Similarly, Chen and Hong [13] used Cesàro sum regularization technique for hypersingularity of dual integral equation.

The generalized weighted -class is generalization of and classes. Therefore, in the present paper, a theorem on degree of approximation of conjugate of signals belonging to the generalized weighted class by product summability means of conjugate series of Fourier series has been established which is in more general form than that of Nigam and Sharma [9]. We also note some errors appearing in the paper of Nigam [8], Nigam and Sharma [9] and rectify the errors pointed out in Remarks 2.2, 2.3 and 2.4.

Let be a -periodic function and integrable in the sense of Lebesgue. The Fourier series of at any point is given by with th partial sum .

The conjugate series of Fourier series (1.1) is given by Let be a given infinite series with sequence of its th partial sums . The transform is defined as the th partial sum of summability, and we denote it by .

If then the infinite series is summable to a definite number , Hardy [14].

If then the infinite series is summable to the definite number by method. The transform of the transform defines product transform and denotes it by . Thus, if then the infinite series is said to be summable by method or summable to a definite number . The is regular method of summability A function , if and , for , if Given a positive increasing function , if Given positive increasing function , an integer , ([2]), if For our convenience to evaluate without error, we redefine the weighted class as follows: If , then the weighted class coincides with the class , we observe that -norm of a function is defined by A signal is approximated by trigonometric polynomials of order , and the degree of approximation is given by Rhoades [3] in terms of , where is a trigonometric polynomial of degree . This method of approximation is called trigonometric Fourier approximation (TFA) [4].

We use the following notations throughout this paper:

2. Previous Result

Nigam [8] has proved a theorem on the degree of approximation of a function , conjugate to a periodic function with period and belonging to the class by product summability means of conjugate series of Fourier series. He has proved the following theorem.

Theorem 2.1 (see [8]). If , conjugate to a -periodic function , belongs to class, then its degree of approximation by product summability means of conjugate series of Fourier series is given by provided satisfies the following conditions: where is an arbitrary number such that , condition (2.2) and (2.3) hold uniformly in and is means of the series (1.2).

Remark 2.2. The proof proceeds by estimating , which is represented in terms of an integral. The domain of integration is divided into two parts—from and . Referring to second integral as , and using Hölder inequality, the author [8] obtains The author then makes the substitution to obtain In the next step is removed from the integrand by replacing it with , while is an increasing function, is now a decreasing function. Therefore, this step is invalid.

Remark 2.3. The proof follows by obtaining , in Theorem 2 of Nigam and Sharma [9], which is expressed in terms of an integral. The domain of integration is divided into two parts—from and . Referring to second integral as , and using Hölder inequality, the authors [9] obtain the following: The authors then make the substitution to get In the next step, is removed from the integrand by replacing it with , while is an increasing function, is now a decreasing function. Therefore, in view of second mean value theorem of integral, this step is invalid.

Remark 2.4. The condition used by Nigam and Sharma [9] is not valid, since as .

3. Main Result

It is well known that the theory of approximation, that is, TFA, which originated from a theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 130 years. These approximations have assumed important new dimensions due to their wide applications in signal analysis [15], in general, and in digital signal processing [16] in particular, in view of the classical Shannon sampling theorem. Broadly speaking, signals are treated as function of one variable and images are represented by functions of two variables.

This has motivated Mittal and Rhoades [17–20] and Mittal et al. [4, 21] to obtain many results on TFA using summability methods without rows of the matrix. In this paper, we prove the following theorem.

Theorem 3.1. If , conjugate to a -periodic function , belongs to the generalized weighted -class, then its degree of approximation by product summability means of conjugate series of Fourier series is given by provided satisfies the following conditions: where is an arbitrary number such that , conditions (3.2) and (3.3) hold uniformly in and is means of the series (1.2) and the conjugate function is defined for almost every by

Note 1. , for .

Note 2. Also for , Theorem 3.1 reduces to Theorem 2.1, and thus generalizes the theorem of Nigam [8]. Also our Theorem 3.1 in the modified form of Theorem 2 of Nigam and Sharma [9].

Note 3. The product transform plays an important role in signal theory as a double digital filter [6] and the theory of machines in mechanical engineering.

4. Lemmas

For the proof of our theorem, the following lemmas are required.

Lemma 4.1. Consider for .

Proof. For and . This completes the proof of Lemma 4.1.

Lemma 4.2. Consider , for and any .

Proof. For . Now considering first term of (4.2) Now considering second term of (4.2) and using Abel’s lemma On combining (4.2), (4.3), and (4.4) This completes the proof of Lemma 4.2.

5. Proof of Theorem

Let denotes the th partial sum of series (1.2). Then following Nigam [8], we have Therefore, using (1.2), the transform of is given by Now denoting transform of as , we write We consider, .

Using Hölder’s inequality Since is a positive increasing function and by using second mean value theorem for integrals, we have Note that , Now, we consider Using Hölder’s inequality Now putting , we have Since is a positive increasing function, so is also a positive increasing function and using second mean value theorem for integrals, we have Combining and yields Now, using the -norm of a function, we get This completes the proof of Theorem 3.1.