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Abstract
Degree of approximation of functions of different classes has been studied by several researchers by different summability methods. In the proposed paper, we have established a new theorem for the approximation of a signal (function) belonging to the -class by
-product summability means of a Fourier series. The result obtained here, generalizes several known theorems.
AMS Subject Classifications:
Public Interest Statement
The theory of summability is a wide field of mathematics as regards to the study of Analysis and Functional Analysis. It has many applications, for instance, in numerical analysis (to speed of the rate of convergence), complex analysis ( for analytic continuums), operator theory, the theory of orthogonal series, and approximation theory, etc.; while the classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. Further, in classical summability theory, the use of infinite matrices has been a significant research area in mathematical analysis as regards to the study of summability of divergent sequences and series. Recently, various summability methods have interesting applications in approximation theory. The approximation of functions by positive linear operators is a significant research area in mathematical analysis with key relevance to studies of computer-aided geometric design and solution of differential equations.
1. Introduction
The theory of summability arose from the process of summation of series and the significance of the concept of summability has been rightly demonstrated in varying contexts, e.g. in Fourier analysis, approximation theory and fixed point theory and many other fields. The theory of approximation of functions has been originated from a well-known theorem of Weierstrass, it has become an exciting interdisciplinary field of study for last 130 years. The approximation of functions by generalized Fourier series, based on trigonometric polynomial is a closely related topic in the recent development of engineering and mathematics. The almost summability method and statistical summability method are now an active area of research in summability theory. The error approximation of periodic functions belonging to various Lipschitz classes through summability method is also an active area of research in the last decades. The engineers and scientist use the properties of approximation of functions for designing digital filters. Psarakis and Moustakides (Citation1997) presented a new -based method for designing Finite Impulse Response digital filters for getting optimum approximation. In similar manner,
-space,
-space, and
-space play an important role for designing digital filters. The approximation of functions belonging to various Lipschitz classes, through trigonometric Fourier approximation using different summability means has been proved by various investigators, like Nigam (Citation2011), Lal (Citation2000), Paikray, Jati, Misra, and Sahoo (Citation2012) and many others. Recently, by generalized H
lders inequality and Minkowski’s inequality, Mishra, Sonavane, and Mishra (Citation2013) have proved
approximation of signals belonging to
-class by
-summability means of conjugate series of Fourier series. Mishra and Sonavane (Citation2015) has proved approximation of functions belonging to the Lipschitz class by product mean
of Fourier series. In an attempt to make an advance study in this direction, in this paper, we obtain a theorem on the approximation of functions belonging to
by
-summability means of Fourier series which generalizes several known and unknown results.
2. Definition and notations
Let be an infinite series with the sequence of partial sum
. Let
and
be sequences of positive real number such that,
and let
The sequence to sequence transformation (Mishra, Palo, Padhy, Samanta, & Misra, Citation2014),
defines the sequence of the
mean of the sequence
generated by the sequence of coefficients
and
.
Similarly, we define the extended Riesz mean,(2.1)
(2.1)
where .
If , then the series
is
summable to s.
Analogous to regularity conditions of Riesz summability (Hardy, Citation1949), we have
(i) |
| ||||
(ii) |
|
defines the sequence of (E, s) mean of the sequence
.
If as
, then
summable to s with respect to (E, s) summability and (E, s) method is regular (Hardy, Citation1949).
Now we define, a new composite transformation over (E, s) of
as
(2.3)
(2.3)
If as
then
is summable to s by
means.
Further as and (E, s) means are regular, so
mean is also regular.
Remark 1
If we put in Equation (2.1) then
-summability method reduces to
-summability and for
it reduces to
-summability.
Let f is a periodic function belonging to
,
, with the partial sum
, then
(2.4)
(2.4)
Here, as regards to Lipschitz classes we may recall that, a signal (function) , if
and , for
, if
Again, , if
where is a positive increasing function.
Similarly, , if
Further as regards to the norm in and
-spaces, we may recall that
-norm of a function
is defined by
and -norm of a function
is defined by
Next, the degree of approximation of a function by a trigonometric polynomial
of order n under
is defined by
and the degree of approximation of of a function
is given by
We use the following notations throughout this paper:
Remark 2
If we take , then
-class coincides with the class
; if
then the class
coincides with
-class and if
then
- class reduces to the
.
3. Known theorems
Dealing with the product (C, 1)(E, q) mean, in Nigam (Citation2011) proved the following theorem.
Theorem 1
If f is a periodic function, Lebesgue integrable on
and belongs to
class, then its degree of approximation is given by
(3.1)
(3.1)
provided satisfies the following conditions:
(3.2)
(3.2)
(3.3)
(3.3)
and(3.4)
(3.4)
where is any arbitrary number such that
, conditions (3.3) and (3.4) hold uniformly in x and
is (C, 1)(E, q) means of the Fourier series (2.4).
Next, dealing with degree of approximation, in Mishra et. al. (Citation2014) proved the following theorem.
Theorem 2
For a positive increasing function and an integer
, if f is a
-periodic function on the class
, then the degree of approximation by product
-summability mean of Fourier series (2.4) is given by
where is
-summability mean.
4. Main theorem
Here, just by replacing Nörlund summability by extended Riesz summability and taking the reverse composition, we have proved the following theorem.
Theorem 3
Let f be a periodic function which is integrable in Lebesgue sense in
. If
class, then its degree of approximation is given by
(4.1)
(4.1)
where is the
transform of
, provided
satisfies the following conditions;
(4.2)
(4.2)
(4.3)
(4.3)
and(4.4)
(4.4)
To prove the theorem, we need the following lemmas.
Lemma 1
, for
Proof
For , as
; so we have
Lemma 2
, for
Proof
For , as
(Jordan’s lemma) and
; so
5. Proof of main theorem
Using Riemann-Lebesgue theorem,(5.1)
(5.1)
As, ,
so, by using Minkowski’s inequality,
Further implies
, thus
Now by Hölder’s inequality and Lemma 1, we have
Also, by 2nd mean value theorem, we have(5.2)
(5.2)
Now by Hlder inequality and Lemma 2, we have
Again by using 2nd mean value theorem(5.3)
(5.3)
Next, by using (5.2) and (5.3), in (5.1) we have
Which completes the proof of theorem.
Corollary 1
If we put in Theorem 3, then the generalized Lipschitz
-class reduces to
, where
is any positive increasing function and
. If f is
-periodic and belonging to class
, then the degree of approximation by
-summability mean of Fourier series is
(5.4)
(5.4)
Corollary 2
If we put and
,
, in Theorem 3, the generalized Lipschitz
-class reduces to
, then the degree of approximation of
periodic function f belonging to class
by
-summability mean of Fourier series is
(5.5)
(5.5)
Corollary 3
If we put ,
,
and
then the generalized Lipschitz
-class reduces to
, then the degree of approximation of
periodic function by
-summability mean of Fourier series
is
(5.6)
(5.6)
Additional information
Funding
Notes on contributors
Tejaswini Pradhan
Tejaswini Pradhan is working as a research scholar in the Department of Mathematics, Veer Surendra Sai University of Technology, Burla, India. Currently she is continuing her PhD work in the field of the Summability theory.
Susanta Kumar Paikray
Susanta Kumar Paikray is currently working as an associate professor in the Department of Mathematics, Veer Surendra Sai University of Technology, Burla, India. He has published more than 35 research papers in various National and International Journals of repute. The research area of Paikray is Summability theory, Fourier series, Operations research and Inventory optimization.
Umakanta Misra
Umakanta Misra is currently working as a professor in the Department of Mathematics, National Institute of Science and Technology, Berhampur, India. He has published more than 130 research papers in various National and International Journals of repute. The research area of Misra is Summability theory, Sequence space, Fourier series, Inventory control, mathematical modeling and Graph Theory.
References
- Hardy, G. H. (1949). Divergent Series (1st ed.). Oxford: Oxford University Press.
- Lal, S. (2000). On degree of approximation of conjugate of a function belonging to weighted (Lp, ξ(t)) class by matrix summability means of conjugate series of a Fourier series. Tamkang Journal of Mathematics, 31, 279–288.
- Mishra, M., Palo, P., Padhy, B. P., Samanta, P., & Misra, U. K. (2014). Approximation of Fourier series of a function of Lipschitz class by product means. Journal of Advances in Mathematics, 9, 2475–2484.
- Mishra, V. N., & Sonovane, V. (2015). Approximation of functions of Lipschitz class by (N, pn) (E, 1) summability means of conjugate series of Fourier series. Journal of Classical Analysis, 6, 137–151.
- Mishra, V. N., Sonavane, V., & Mishra, L. N. (2013). Lr-Approximation of signals (functions) belonging to weighted (Lp, ξ(t))-class by (C1 Np)-summability method of conjugate series of its Fourier series. Journal of Inequality and Applications, Article ID, 440, 1–15. doi:10.1186/1029-242X-2013-440
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- Paikray, S. K., Jati, R. K., Misra, U. K., & Sahoo, N. C. (2012). On degree of approximation of Fourier series by product means. General Math. Notes, 13, 22–30.
- Psarakis, E. Z., & Moustakides, G. V. (1997). An (L2)-based method for the design of 1-D zero phase FIR digital filters. IEEE Transactions on Circuits and Systems, I44, 551–601.
- Hardy, G.H. (1949). Divergent series. First EditionOxford University Press.