Trigonometric Approximation of Signals (Functions) Belonging to the Lip(ξ(t),r),(r＞1)-Class by (E,q) (q＞0)-Means of the Conjugate Series of Its Fourier Series ()
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 En(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 function
is 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 nth 
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 nth 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.
NOTES