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 ![](https://www.scirp.org/html/7-5300266\21fe4c3e-0528-4d3a-a5e5-3d4b1a4a4cd5.jpg)
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 ![](https://www.scirp.org/html/7-5300266\ba538ce0-b491-4500-9af3-eae8258c9512.jpg)
, if
(1.4)
We observe that
(1.5)
The
-norm of a signal
is defined by
![](https://www.scirp.org/html/7-5300266\bf455e66-5059-457e-8238-ec901666b2ac.jpg)
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].
![](https://www.scirp.org/html/7-5300266\3fb9b7bc-eb45-40cb-a20e-1a95cef9b27a.jpg)
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
![](https://www.scirp.org/html/7-5300266\03c35ab1-60e9-44db-9203-942cf6da70b5.jpg)
.
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
![](https://www.scirp.org/html/7-5300266\a605fa3a-42d7-427c-8610-c025c0a71f8d.jpg)
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
![](https://www.scirp.org/html/7-5300266\fba5207c-7e3f-4b4d-ae4a-21cdaf577785.jpg)
5. Proof of Theorem 3.1
Let
denote the partial sum of series (1.9), then we have
![](https://www.scirp.org/html/7-5300266\faef77cd-ce8b-4f9a-a05b-fbb27744c812.jpg)
Therefore the
transform
of
is given by
![](https://www.scirp.org/html/7-5300266\772fc6b2-7e93-4fef-bb9f-7528203ebc60.jpg)
(5.1)
Now, we consider
![](https://www.scirp.org/html/7-5300266\5b0e241e-6945-4f37-bed3-f2c645cada98.jpg)
Applying Hölder’s inequality, using the fact that
due to
condition (3.2) and Lemma 4.1, we have
![](https://www.scirp.org/html/7-5300266\191eab2d-afda-4cef-b91f-d9fec4cd85db.jpg)
Since
is positive increasing function so using condition (3.4), we have
![](https://www.scirp.org/html/7-5300266\5ac5207d-484b-4025-ae0f-24c70b5fac9d.jpg)
and Second Mean Value Theorem for integrals, we get
![](https://www.scirp.org/html/7-5300266\622b64c1-c9fd-4cf1-8c54-eae3331d361f.jpg)
(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
![](https://www.scirp.org/html/7-5300266\3daa89f6-46c4-4c38-a9bf-b8b50e838bd0.jpg)
(5.3)
in view of increasing nature of
, ![](https://www.scirp.org/html/7-5300266\a86887e1-84e6-4061-850c-43e31b43d978.jpg)
where
lie in
, Second Mean Value Theorem for integrals and Note 3.2.
Collecting (5.1) - (5.3), we get
![](https://www.scirp.org/html/7-5300266\866cd611-a64c-4e65-baa5-eb9cef7310e5.jpg)
Now, using the
-norm of a function, we get
![](https://www.scirp.org/html/7-5300266\b2843dc9-08f8-4cab-bd2d-83196e9202c7.jpg)
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
![](https://www.scirp.org/html/7-5300266\189250be-f466-4a43-961a-1dfbab0a9769.jpg)
or,
![](https://www.scirp.org/html/7-5300266\d0c936a9-664d-4e54-afd8-4538882b90d1.jpg)
or,
![](https://www.scirp.org/html/7-5300266\ca2e6caa-d1cb-4fbf-94a7-409288e9a744.jpg)
For if not the right hand side of the above equation will be O(1), therefore, we have
![](https://www.scirp.org/html/7-5300266\9d5fb019-fb4d-473a-8eb4-6e41b572a7c4.jpg)
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
![](https://www.scirp.org/html/7-5300266\64ceeb80-e676-4fe3-a4c1-3faeb75e5b69.jpg)
Proof. For
in Corollary 6.1, we get
![](https://www.scirp.org/html/7-5300266\4161a9aa-83fc-4fcf-8a9b-96e8fbb52e95.jpg)
Thus, we have
![](https://www.scirp.org/html/7-5300266\1e76cf5b-f101-4ef3-8341-5c77fbef32e2.jpg)
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
![](https://www.scirp.org/html/7-5300266\28617cda-70cc-4137-b400-12c96795067d.jpg)
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
![](https://www.scirp.org/html/7-5300266\9ff35749-8f73-475b-a962-05f3e1bc2076.jpg)
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