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Trigonometric approximation of functions belonging to Lipschitz class by matrix () operator of conjugate series of Fourier series
Advances in Difference Equations volume 2013, Article number: 127 (2013)
In the present paper, a new theorem on the degree of approximation of a function , conjugate to a 2π periodic function f belonging to the Lipα () class without the monotonicity condition on the generating sequence has been established, which in turn generalizes the results of Lal (Appl. Math. Comput. 209: 346-350, 2009) on a Fourier series.
MSC:40G05, 41A10, 42B05, 42B08.
The degree of approximation of functions belonging to Lipα, , and , -classes through trigonometric Fourier approximation using different summability matrices with monotone rows has been proved by various investigators like Khan , Mittal et al. [2, 3], Mittal, Rhoades and Mishra , Qureshi , Chandra , Leindler , Rhoades et al. . Recently Lal  has proved a theorem on the degree of approximation of a function f belonging to the Lipα () class by summability method of its Fourier series. Lal  has assumed monotonicity on the generating sequence . The approximation of a function , conjugate to a 2π periodic function to () using product ()-summability has not been studied so far. In this paper, we obtain a new theorem on the degree of approximation of a function , conjugate to a 2π periodic function () class without monotonicity condition on the generating sequence .
Let be a given infinite series with the sequence of n th partial sums . Let be a non-negative sequence of constants, real (R) or complex, and let us write
The sequence to sequence transformation defines the sequence of Nörlund means of the sequence , generated by the sequence of coefficients . The series is said to be summable to the sum s if exists and is equal to a finite number s. In the special case, in which
the Nörlund summability reduces to the familiar summability.
The product of summability with a summability defines summability. Thus the mean is given by .
If as , then the infinite series or the sequence is said to be summable to the sum s if exists and is equal to s.
Let be a 2π-periodic function and Lebesgue integrable. The Fourier series of is given by
with th partial sum called the trigonometric polynomial of degree (order) n of the first () terms of the Fourier series of f.
The conjugate series of Fourier series (1.1) is given by
A function if
-norm of a function is defined by .
The degree of approximation of a function by the trigonometric polynomial of order n under the sup norm is defined by 
and of a function is given by .
The conjugate function is defined for almost every x by
We note that and are also trigonometric polynomials of degree (or order) n.
Abel’s transformation: The formula
where , , if , , which can be verified, is known as Abel’s transformation and will be used extensively in what follows.
If are non-negative and non-increasing, the left-hand side of (1.3) does not exceed in absolute value. In fact,
We write throughout the paper
, where τ denotes the greatest integer not exceeding , , .
2 Known results
In a recent paper Lal  obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using Cesàro-Nörlund -summability means of its Fourier series with non-increasing weights . He proved the following theorem.
Theorem 2.1 Let be a regular Nörlund method defined by a sequence such that
Let be a 2π-periodic function belonging to Lipα (), then the degree of approximation of f by means of its Fourier series (1.1) is given by
Remark 1 In the proof of Theorem 2.1 of Lal [5, p.349], the estimate for the case is obtained as
Since , the e is not needed in (2.2) for the case (cf. [, p.6870]).
Remark 2 Lal  has used the monotonicity condition on the generating sequence in the proof of Theorem 2.1 but has not mentioned it in the statement.
3 Main theorem
The theory of approximation is a very extensive field and the study of theory of trigonometric approximation is of great mathematical interest and of great practical importance. It is well known that the theory of approximations, i.e., TFA, which originated from a well-known theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 130 years. These approximations have assumed important new dimensions due to their wide applications in signal analysis  in general and in digital signal processing  in particular, in view of the classical Shannon sampling theorem. Mittal et al. [2–4, 14] have obtained many interesting results on TFA using summability methods without monotonicity on the rows of the matrix T: a digital filter. Broadly speaking, signals are treated as functions of one variable and images are represented by functions of two variables. But till now, nothing seems to have been done so far to obtain the degree of approximation of conjugate of a function using product summability method of its conjugate series of a Fourier series. The observations of Remarks 1 and 2 motivated us to determine a proper set of conditions to prove Theorem 2.1 on the conjugate series of its Fourier series. The series, conjugate to a Fourier series, is not necessarily a Fourier series. Hence a separate study of conjugate series is desirable, which attracted the attention of researchers.
Therefore, the purpose of present paper is to establish a quite new theorem on the degree of approximation of a function , conjugate to a 2π-periodic function f belonging to the Lipα () class by means of conjugate series of its Fourier series without monotonicity on the generating sequence (that is, weakening the conditions on the filter, we improve the quality of a digital filter [, p.4485]). More precisely, we prove the following theorem.
Theorem 3.1 Let be the regular Nörlund summability matrix generated by the non-negative such that
Let be a 2π-periodic signal (function). Then the degree of approximation of , conjugate to () by means of conjugate series of its Fourier series, is given by
Remark 3 For a non-increasing sequence , we get
Thus the condition (3.1) holds for a non-increasing sequence . Hence our Theorem 3.1 generalizes Theorem 2.1 on conjugate series of its Fourier series.
Note 1 The product transform plays an important role in signal theory as a double digital filter .
We need the following lemmas for the proof of our theorem.
Lemma 1 If is positive and , then for , and for any n, we have
Proof Let . Then
and, by (1.4), we have
Since and , we have
and, in case , we would have
This completes the proof of Lemma 1. □
Lemma 2 for .
Proof For , and .
This completes the proof of Lemma 2. □
Lemma 3 Let be a non-negative sequence satisfying (3.1), then
Proof For , we have
in view of , for .
Again, using , for and changing the order of summation, we find
Using Lemma 1, we have
Using Abel’s transformation, we obtain
Therefore, we have
by virtue of the fact that , , and .
On combining (4.3) to (4.5), we get
in view of (3.1) and .
Finally, collecting (4.1), (4.2) and (4.6) yields Lemma 3.
This completes the proof of Lemma 3. □
5 Proof of the theorem
Let denote the partial sum of series (1.2), then we have
Denoting means of by , we write
If , then
Now, using Lemma 2, we have
Using Lemma 3, we obtain
On combining (5.1) with (5.5) and using the inequality , for higher values of n, we have
This completes the proof of Theorem 3.1.
Several results concerning the degree of approximation of periodic signals (functions) belonging to the Lipschitz class by Matrix Operator have been reviewed and the condition of monotonicity on the generating sequence has been relaxed. Further, a proper set of conditions has been discussed to rectify the errors and applications pointed out in Remarks 1 and 2. Some interesting applications of the operator used in this paper were pointed out in Note 1.
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Dedicated to Professor Hari M Srivastava.
This research work is supported by CPDA, SVNIT, Surat, India. The authors thank the anonymous reviewers for their valuable suggestions, which substantially improved the standard of the paper. Special thanks are due to Prof. Hari Mohan Srivastava, for kind cooperation, smooth behavior during communication and for his efforts to send the reports of the manuscript timely.
The authors declare that they have no competing interests.
LNM computed lemmas and established the main theorem in this direction. LNM and VNM conceived of the study and participated in its design and coordination. LNM, VNM contributed equally and significantly in writing this paper. All the authors drafted the manuscript, read and approved the final manuscript.
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Mishra, L.N., Mishra, V.N. & Sonavane, V. Trigonometric approximation of functions belonging to Lipschitz class by matrix () operator of conjugate series of Fourier series. Adv Differ Equ 2013, 127 (2013). https://doi.org/10.1186/1687-1847-2013-127
- conjugate Fourier series
- () class
- degree of approximation
- product summability transform