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Approximation of functions belonging to class by summability of conjugate series of Fourier series
Journal of Inequalities and Applications volume 2012, Article number: 296 (2012)
In this paper, a new theorem concerning the degree of approximation of the conjugate of a function belonging to class by summability of its conjugate series of a Fourier series has been proved. Here the product of Euler summability method and Nörlund method has been taken.
MSC:42B05, 42B08, 40G05.
Khan [1, 2] has studied the degree of approximation of a function belonging to -class by Nörlund means. Generalizing the results of Khan [1, 2], many interesting results have been proved by various investigators like Mittal et al. [3–5], Mittal, Rhoades and Mishra , Mittal and Singh , Rhoades et al. , Mishra et al. [9, 10] and Mishra and Mishra  for functions of various classes Lipα, , and , () by using various summability methods. 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 Fourier series. In this paper, we obtain a new theorem on the degree of approximation of a function , conjugate to a periodic function -class, by product summability means.
Let be a given infinite series with the sequence of its n th partial sums . Let be a non-negative generating sequence of constants, real or complex, and let us write
The conditions for regularity of Nörlund summability are easily seen to be
, as .
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 s.
The transform is defined as the n th partial sum of summability, and we denote it by . If
then the infinite series is said to be summable to the sum s Hardy . The transform of the transform defines product transform and denotes it by . This is if
If as , then the infinite series is said to be summable to the sum s.
A function if
and , for if
Given a positive increasing function , ,  if
we observe that
-norm of a function is defined by
-norm of a function is defined by .
A signal (function) f is approximated by trigonometric polynomials of order n and the degree of approximation is given by Zygmund 
in terms of n, where is a trigonometric polynomial of degree n. This method of approximation is called Trigonometric Fourier Approximation (TFA) .
The degree of approximation of a function by a trigonometric polynomial of order n under sup norm is defined by
Let be a 2π-periodic function and Lebesgue integrable. The Fourier series of is given by
with n th partial sum .
The conjugate series of Fourier series (1.7) is given by
means reduces to means if .
means reduces to means if and ∀n.
means reduces to means if ∀n.
means reduces to means if , .
means reduces to means if , and ∀n.
means reduces to means if and ∀n.
We use the following notations throughout this paper:
2 Main result
The approximation of a function , conjugate to a periodic function using product summability, has not been studied so far. Therefore, the purpose of the 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 -class, by means of conjugate series of Fourier series. In fact, we prove the following theorem.
Theorem 2.1 If is conjugate to a 2π-periodic function f belonging to -class, then its degree of approximation by product summability means of conjugate series of Fourier series is given by
provided satisfies the following conditions:
where δ is an arbitrary number such that , , , conditions (2.2) and (2.3) hold uniformly in x and is means of the series (1.8), and the conjugate function is defined for almost every x by
Note 2.2 , for .
Note 2.3 The product transform plays an important role in signal theory as a double digital filter  and the theory of machines in mechanical engineering.
For the proof of our theorem, the following lemmas are required.
Lemma 3.1 for .
Proof For , and ,
This completes the proof of Lemma 3.1. □
Lemma 3.2 for and any n.
Proof For , ,
Now, considering the first term of equation (3.1),
Now, considering the second term of equation (3.1) and using Abel’s lemma
On combining (3.1), (3.2) and (3.3), we have
This completes the proof of Lemma 3.2. □
4 Proof of theorem
Let denote the partial sum of series (1.8), we have
Therefore, using (1.2), the transform of is given by
Now, denoting transform of as , we write
Using Hölder’s inequality, equation (2.2) and Lemma (3.1), we get
Since is a positive increasing function, using the second mean value theorem for integrals,
Now, we consider
Using Hölder’s inequality, equation (3.2) and Lemma 3.2, we have
Now, putting ,
Since is a positive increasing function, so is also a positive increasing function and using the second mean value theorem for integrals, we have
Combining and yields
Now, using the -norm of a function, we get
This completes the proof of Theorem 2.1.
The study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. The following corollaries can be derived from our main Theorem 2.1.
Corollary 5.1 If , , then the class , reduces to the class , and the degree of approximation of a function , conjugate to a 2π-periodic function f belonging to the class , by -means is given by
Proof We have
Thus, we get
This completes the proof of Corollary 5.1. □
Corollary 5.2 If for and in Corollary 5.1, then and
Proof For , we get
Thus, we get
This completes the proof of Corollary 5.2. □
Corollary 5.3 If , , then the class , , reduces to the class , and if , then summability reduces to summability and the degree of approximation of a function , conjugate to a 2π-periodic function f belonging to the class , by -means is given by
Corollary 5.4 If for and in Corollary 5.3, then and
Remark An independent proof of above Corollary 5.3 can be obtained along the same line of our main theorem.
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The authors take this opportunity to express their gratitude to prof. Huzoor Hasan Khan, Department of Mathematics, Aligarh Muslim University, Aligarh, for suggesting the problem as well as his valuable suggestions for the improvement of the paper. The authors would like to thank the anonymous referees for several useful interesting comments about the paper. The authors are also thankful to Prof. Ravi P. Agrawal, editor-in-chief of JIA, Texas A & M University, Kingsville. Special thanks are due to our great master and friend academician Prof. Vijay Gupta, Netaji Subhas Institute of Technology, New Delhi, 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.
VNM, KK and LNM contributed equally to this work. All the authors read and approved the final manuscript.
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Mishra, V.N., Khatri, K. & Mishra, L.N. Approximation of functions belonging to class by summability of conjugate series of Fourier series. J Inequal Appl 2012, 296 (2012). https://doi.org/10.1186/1029-242X-2012-296
- degree of approximation
- -class of function
- product summability
- conjugate Fourier series
- Lebesgue integral