ABSTRACT

Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades et al. [14] have determined the degree of approximation of 2π-periodic signals (functions) belonging to various classes , and of functions through trigonometric Fourier approximation (TFA) using different summability matrices with monotone rows. Recently, Mittal et al. [15], Mishra and Mishra [16], Mishra [17] have obtained the degree of approximation of signals belonging to -class by general summability matrix, which generalizes the results of Leindler [8] and some of the results of Chandra [7] by dropping monotonicity on the elements of the matrix rows (that is, weakening the conditions on the filter, we improve the quality of digital filter). In this paper, a theorem concerning the degree of approximation of the conjugate of a signal (function) f belonging to class by summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Chandra [7] and Shukla [18].

1. Introduction

The theory of approximation is a very extensive field and the study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. Broadly speaking, Signals are treated as functions of one variable and images are represented by functions of two variables. The study of these concepts is directly related to the emerging area of information technology. Khan [1-4] and Mittal, Rhoades and Mishra [12] have initiated the studies of error estimates E_n(f) through trigonometric Fourier approximation (TFA) using different summability matrices. Chandra [7] has studied the degree of approximation of a signal (function) belonging to Lip α-class by (E,q) means, q > 0.

Generalizing the result of Chandra [7], very interesting result has been proved by Shukla [18] for the signals (functions) of -class through trigonometric Fourier approximation by applying (E,q) (q > 0) summability matrix.

Let be a given infinite series with sequence of its partial sums.

The transform is defined as the partial sum of summability and we denote it by

If

as,  (1.1)

then the series is said to be summable to a definite number “s” [19].

A signal (function) if

(1.2)

and [1], if

(1.3)

Given a positive increasing function

, if

(1.4)

We observe that

(1.5)

The -norm of a signal is defined by

The -norm of a signal is defined by

. (1.6)

The degree of approximation of a function by trigonometric polynomial of order “” under sup norm is defined by Zygmund [20].

and of a functionis given by

(1.7)

in terms of n, where is a trigonometric polynomials of order “n”.

This method of approximation is called Trigonometric Fourier Approximation (TFA) [12].

Let be a -periodic signal (function) and Lebesgue integrable. The Fourier series of is given by

(1.8)

with partial sum called trigonometric polynomial of degree (order) n of the first (n + 1) terms of the Fourier series of f.

The conjugate series of Fourier series (1.8) is given by

. (1.9)

We note that is also trigonometric polynomial of degree (or order) “n”.

We use the following notations throughout this paper

.

2. Known Results

Chandra [7] has studied the degree of approximation to a function by of Fourier series (1.8) by proving the following theorem. He proved:

Theorem 2.1 The degree of approximation of a periodic function f(x) with period and belonging to the class by Euler’s mean of its Fourier series is given by

(2.1)

where is the Euler mean of order q > 0 of the sequence of partial sums of the Fourier series (1.8) of the function f at a point x in.

Shukla [18] improved Theorem 2.1 by extending to a function by matrix means of the conjugate series (1.9) of its Fourier series (1.8). He proved:

Theorem 2.2 Let, , be a -periodic and Lebesgue integrable function of “t” in the interval. If

(2.2)

and

(2.3)

where is an arbitrary number such that, s being conjugate to with, then the degree of approximation of the conjugate to a function, by means, , of the conjugate series (1.9) of its Fourier series (1.8) will be given by

(2.4)

where is n^th mean of the sequence of partial sums of the conjugate series (1.9) of the Fourier series (1.8) of the function f at every point x in at which

(2.5)

exists.

3. Main Result

The purpose of the present paper is to extend Theorems 2.1 and 2.2 on the degree of approximation of signal conjugate to a 2π-periodic signal

class by summability means with a proper set of conditions. More precisely, we prove:

Theorem 3.1

If conjugate to a 2π-periodic signal (function) f belonging to -class, then its degree of approximation by means of conjugate series of Fourier series (1.9) is given by

(3.1)

provided positive increasing ξ(t) satisfies the following conditions

(3.2)

(3.3)

and

is non-increasing in “t”,     (3.4)

where is an arbitrary number such that

, , , condition (3.2) and (3.3) hold uniformly in x and is the nth means of the series (1.9) and the conjugate function is defined for almost every x by

(3.5)

Note 3.2 Using condition (3.4), we get

Note 3.3 Also, if, then our main Theorem (3.1) reduces to Theorem 2.2, and thus generalizes the theorem of Shukla [18].

Note 3.4 The transform (E, q) plays an important role in signal theory and the theory of Machines in Mechanical Engineering.

4. Lemma

For the proof of our theorem, we need the following lemma.

Lemma 4.1 [18]: For we have

5. Proof of Theorem 3.1

Let denote the partial sum of series (1.9), then we have

Therefore the transform of is given by

 (5.1)

Now, we consider

Applying Hölder’s inequality, using the fact that

due to condition (3.2) and Lemma 4.1, we have

Since is positive increasing function so using condition (3.4), we have

and Second Mean Value Theorem for integrals, we get

(5.2)

Now, we consider

.

Again applying Hölder’s inequality, using the fact that

due to condition (3.3) and Lemma 4.1, we obtain

(5.3)

in view of increasing nature of,

where lie in, Second Mean Value Theorem for integrals and Note 3.2.

Collecting (5.1) - (5.3), we get

Now, using the -norm of a function, we get

This completes the proof of Theorem 3.1.

6. Corollaries

The following corollaries can be derived form Theorem 3.1.

Corollary 6.1: If then the class

, reduces to the class,

and the degree of approximation of a function, conjugate to a -periodic function f belonging to the class is given by

(6.1)

Proof. Putting in Theorem 3.1, we have

or,

or,

For if not the right hand side of the above equation will be O(1), therefore, we have

This completes the proof of Corollary 6.1.

Corollary 6.2 If for and in Theorem 3.1, then In this case, the degree of approximation of a function, conjugate to a -periodic function f belonging to the class is given by

Proof. For in Corollary 6.1, we get

Thus, we have

This completes the proof of Corollary 6.2.

7. An Example

Consider an infinite series

(7.1)

The n^th partial sums of series (7.1) at is given by

Since does not exist. Therefore the series (7.1)

is non-convergent.

Now, we have the (E,q) transform of (7.1) is given by

Here, does not exist. Hence the series (7.1) is not summable, while the series (7.1) is product summable.

8. Conclusion

Several results concerning to the degree of approximation of periodic signals (functions) belonging to the Lipschitz class by matrix (E,q) operator have been reviewed. Further, a proper set of conditions have been discussed to rectify the errors. Some interesting application of the operator (E,q) used in this paper pointed out in Note 3.4. An example has been discussed also.

9. Acknowledgements

The authors are very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the presentation of the paper considerably. The authors are also thankful to all the members of editorial board of Advances in Pure Mathematics (APM) and Dr. Melody Liu, APM Editorial Board Assistant for their kind cooperation and smooth behavior during communication.

REFERENCES

NOTES

^*The authors declare that they have no competing interests. All the authors contributed equally to this work and they read and approved the final manuscript.