Trigonometric Approximation of Signals ( Functions ) Belonging to the , , Lip t r 1 r-Class by , q , 0 E q-Means of the Conjugate Series of Its Fourier Series *

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 Lip   , Lip r   Lip     , r t  , and W L 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    , t r    , Lip r  -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     , Lip t r    , E q     , Lip t r    , E q 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].

 -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

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][2][3][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] q n E  sum of summability and we denote it by If then the series is said to be summable to a definite number "s" [19].
A signal (function Given a positive increasing function We observe that The -norm of a signal The degree of approximation of a function f R R  n by trigonometric polynomial n of order " " under sup norm t  is defined by Zygmund [20].
in terms of n, where is a trigonometric polynomials of order "n".
This method of approximation is called Trigonometric Fourier Approximation (TFA) [ 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 n is also trigonometric polynomial of degree (or order) "n".
We use the following notations throughout this paper

Known Results
Chandra [7] has studied the degree of approximation to a function 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

 
where q T x th n n is the Euler mean of order q > 0 of the sequence   n s of partial sums of the Fourier series matrix means of the conjugate series (1.9) of its Fourier series (1.8).He proved: , s being conjugate to with , then the degree of approximation of the conjugate to a function means, , of the conjugate series (1.9) of its Fourier series (1.8) will be given by 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 (2.5) exists.

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 , provided positive increasing ξ(t) satisfies the following conditions where  is an arbitrary number such that   , condition (3.2) and (3.3) hold uniformly in x and is the nth E q means of the series (1.9) and the conjugate function , 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.

Lemma
For the proof of our theorem, we need the following lemma.
Lemma 4.1 [18]: denote the partial sum of series (1.9), then we have Applying Hölder's inequality, using the fact that is positive increasing function so using condition (3.4), we have and Second Mean Value Theorem for integrals, we get Again applying Hölder's inequality, using the fact that 3) and Lemma 4.1, we obtain in view of increasing nature of   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.

Corollaries
The following corollaries can be derived form Theorem 3.1. . 1

  ,
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.
This completes the proof of Corollary 6.2.

An Example
Consider an infinite series    The n th partial sums s of series (7.1) at 0 x  is given by is non-convergent.Now, we have the (E,q) transform of (7.1) is given by q s k q n q k q q q q q q q q q q n n E  Here, does not exist.Hence the series (7.1) is not summable, while the series (7.1) is product summable.

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.
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 . Therefore the series(7.1) 1.