On trigonometric approximation of () function by product means of its Fourier series
© Mishra et al.; licensee Springer 2013
Received: 18 January 2013
Accepted: 28 May 2013
Published: 25 June 2013
In the present paper, we generalize a theorem of Lal and Singh (Indian J. Pure Appl. Math. 33(9):1443-1449, 2002) on the degree of approximation of a function belonging to the weighted ()-class using product means of its Fourier series. We have used here the modified definition of the weighted ()-class of functions in view of Khan (Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 31:123-127, 1982) and rectified some errors appearing in the paper of Lal and Singh (Indian J. Pure Appl. Math. 33(9):1443-1449, 2002). A few applications of approximation of functions will also be highlighted.
MSC: 40C99, 40G99, 41A10, 42B05, 42B08.
Dedicated to Professor Hari M Srivastava
Approximation by trigonometric polynomials is at the heart of approximation theory. The most important trigonometric polynomials used in the approximation theory are obtained by linear summation methods of Fourier series of 2π-periodic functions on the real line (i.e. Cesàro means, Nörlund means, Euler means and Product Cesàro-Nörlund means, Cesàro-Euler means etc.). Much of the advance in the theory of trigonometric approximation is due to the periodicity of the functions. Various investigators such as Khan [1–3], Qureshi , Chandra , Leindler , Mittal et al. , Mittal, Rhoades and Mishra , Mishra , Rhoades et al.  have determined the degree of approximation of 2π-periodic functions belonging to different classes Lipα, , and of functions through trigonometric Fourier approximation (TFA) using different summability matrices. Recently, Mittal et al.  have obtained the degree of approximation of functions belonging to the -class by a general summability matrix, which generalizes the results of Chandra . In this paper, we determine the degree of approximation of functions belonging to the ()-class by using means of its Fourier series, which in turn generalizes the result of Lal and Singh . We also note some errors appearing in the paper of Lal and Singh  and rectify them in the light of observations of Khan .
with n th partial sum called trigonometric polynomial of degree (order) n of the first terms of the Fourier series of f.
We note that if , then the weighted class coincides with the class and if , then the class coincides with the class . The class for .
then an infinite series with the partial sums is said to be summable to the definite number s (Hardy ).
The transform of the transform defines the transform of the partial sums of the series , i.e., the product summability is obtained by superimposing summability on summability.
We note that and are also trigonometric polynomials of degree (or order) n.
The product transform plays an important role in signal theory as a double digital filter.
The Riemann-Lebesgue theorem states that if is integrable over , then as , we have and .
2 Known theorem
Lal and Singh  have obtained a theorem on the degree of approximation of a function belonging to the class by means of its Fourier series. They proved the following theorem.
where δ is an arbitrary number such that , conditions (2.1) and (2.2) hold uniformly in x and are means of series (1.1).
In the next step is removed from the integrand by replacing it with . While is an increasing function, is now a decreasing function. Therefore, from the second mean value theorem of integrals, this step is invalid.
note that . Therefore one has , which need not be since ϵ might be for some .
3 Main result
The observation of Remark 1 motivated us to determine a proper set of conditions to extend Theorem 2.1 on the degree of approximation of functions f of the weighted ()-class by product means of its Fourier series. More precisely, we prove the following.
where δ is an arbitrary number such that , q is the conjugate index of p, , , conditions (3.2), (3.3) hold uniformly in x, and are means of Fourier series (1.1).
Note 1 Using condition (3.4), we get for .
Note 2 Conditions (2.1) and (2.2) of Theorem 2.1 and (3.2) and (3.3) of Theorem 3.1 are derived from the theorem of Khan .
For the proof of our theorem, we need the following lemma.
This completes the proof of Lemma 4.1. □
5 Proof of Theorem 3.1
in view of increasing nature of , , , where lie in .
This completes the proof of Theorem 3.1.
6 Corollaries and example
The following corollaries can be derived from Theorem 3.1.
This completes the proof of Corollary 1. □
since otherwise the right-hand side of the above equation will not be .
This completes the proof of Corollary 2. □
This completes the proof of Corollary 3. □
Also, does not exist. Therefore the series (6.1) is not summable.
Here does not exist, the series (6.1) is not summable.
the series (6.1) is summable.
Therefore the series (6.1) is neither summable nor summable. But it is summable to 0. Therefore the product summability is more powerful than and . Consequently, gives better approximation than the individual methods and .
(ii) If we take the sequence for , then there are two cases:
(a) If , then the sequence is already convergent and so it is summable.
(b) If , then the sequence is not summable but summable.
(iii) It is a well-known result that a Hausdorff matrix is a Nörlund matrix if and only if it is a Cesàro matrix. Cesàro means and Euler means both are Hausdorff means, but they are not comparable. Two matrices are called comparable if either they are equivalent (that is, they sum the same set of sequences), or one method is stronger than the other (that is, it has the larger convergence domain). Hardy , in his book on Divergent Series, showed that and methods are not comparable.
is bounded but not summable.
Several results concerning the degree of approximation of periodic functions belonging to the generalized weighted ()-class by product means of its Fourier series have been reviewed. Further, a proper set of conditions have been discussed to rectify the errors pointed out in Remarks 1 and 2. The theorem of this paper is an attempt to formulate the problem of approximation of the function () through trigonometric polynomials generated by the product summability transform of the Fourier series of f in a simpler manner. The product summability used in this paper plays an important role in signal theory as a double digital filter and the theory of machines in mechanical engineering.
The authors would like to thank the anonymous referees for several useful interesting comments and suggestions about the paper. Special thanks are to Professor Hari Mohan Srivastava for kind cooperation, smooth behavior during communication and for his efforts to send the reports of the manuscript timely. This research work is supported by CPDA, SVNIT, Surat (Gujarat), India.
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