Abstract
A new estimate for the degree of approximation of a function class by means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001). Some corollaries have also been deduced from our main theorem.
1. Introduction
The degree of approximation of a function belonging to various classes using different summability method has been determined by several investigators like Khan [1, 2], V. N. Mishra and L. N. Mishra [3], Mishra et al. [4–6], and Mishra [7, 8]. Summability techniques were also applied on some engineering problems; for example, Chen and Jeng [9] implemented the Cesàro sum of order and , in order to accelerate the convergence rate to deal with the Gibbs phenomenon, for the dynamic response of a finite elastic body subjected to boundary traction. Chen and Hong [10] used Cesàro sum regularization technique for hyper singularity of dual integral equation. Summability of Fourier series is useful for engineering analysis, for example, [11]. Recently, Mursaleen and Mohiuddine [12] discussed convergence methods for double sequences and their applications in various fields. In sequel, Alexits [13] studied the degree of approximation of the functions in class by the Cesàro means of their Fourier series in the sup-norm. Chandra ([14, 15]), Mohapatra and Chandra ([16, 17]), and Szal [18] have studied the approximation of functions in Hölder metric. Mishra et al. [6] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals. In 2008, Singh and Mahajan [19] studied error bound of periodic signals in the Hölder metric and generalized the result of Lal and Yadav [20] under much more general assumptions. Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. Especially, Psarakis and Moustakides [21] presented a new based method for designing the finite impulse response (FIR) digital filters and got corresponding optimum approximations having improved performance. We also discuss an example when the Fourier series of the signal has Gibbs phenomenon.
For a -periodic signal , periodic integrable in the sense of Lebesgue. Then the Fourier series of is given by with th partial sum called trigonometric polynomial of degree (or order) and given by The conjugate series of Fourier series (1) is given by with th partial sum .
Let and denote two given moduli of continuity such that Let denote the Banach space of all -periodic continuous functions defined on under the sup-norm. The space where includes the space . For some positive constant , the function space is defined by with norm defined by where and are increasing functions of , with the understanding that . If there exist positive constants and such that and , then the space is Banach space [22] and the metric induced by the norm on is said to be Hölder metric. Clearly is a Banach space which decreases as increases; that is, Let be a given infinite series with the sequence of th partial sums . Let be a nonnegative sequence of constants, real 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 if exists and is equal to .
In the special case in which then Nörlund summability reduces to the familiar summability.
An infinite series is said to be summable to if The transform is defined as the th partial sum of summability and we denote it by .
If then the infinite series is said to be summable to .
The transform of the transform defines the transform of the partial sums of the series ; that is, the product summability is obtained by superimposing summability on summability. Thus, if where denotes the transform of , then the series with the partial sums is said to be summable to the definite number and we can write The transform of the transform defines product transform and denote it by . If then the infinite series is said to be summable to .
We note that , and are also trigonometric polynomials of degree (or order) .
The relation between Cesàro mean and Fejér mean is given by where the Fejér mean is and Cesàro mean is Some important particular cases are as follows:(1) means, when .(2) means, , where
Remark 1. If we take , then reduces to class.
The conjugate function is defined for almost every by see [23, page 131].
We write throughout the paper We note that the series, conjugate to a Fourier series, is not necessarily a Fourier series [23, 24]. Hence, a separate study of conjugate series is desirable and attracted the attention of researchers.
2. Known Results
In 2001, Lal and Yadav [20] established the following theorem to estimate the error between the input signal and the signal obtained after passing through the - transform.
Theorem 2 (see [20]). If a function is -periodic function and belongs to class Lip , , then degree of approximation by means of its Fourier series is given by Recently, Singh and Mahajan [19] generalized the above result under more general assumptions. They proved the following.
Theorem 3 (see [19]). Let defined in (5) be such that then, for and , we have
Theorem 4 (see [19]). Consider defined in (5) and for and , we have
3. Main Theorem
In this paper, we prove a theorem on the degree of approximation of a function , conjugate to a -periodic function belonging to class by means of conjugate series of its Fourier series. This work generalizes the results of Singh and Mahajan [19] on summability of conjugate Fourier series. We will measure the error between the input signal and the processed output signal , by establishing the following theorems.
Theorem 5. The functions satisfy the following conditions: where and are increasing functions of . Let be the Nörlund summability matrix generated by the nonnegative such that Then, for , we have and if satisfies (28), then for , we have
Remark 6. The product transform plays an important role in signal theory as a double digital filter and theory of Machines in Mechanical Engineering [4, 5].
4. Lemmas
In order to prove our main Theorem 5, we require the following lemmas.
Lemma 7. If , then for , we have It is easy to verify the following.
Lemma 8. Let be a nonnegative and nonincreasing sequence satisfies (30); then for , we have
Proof. Consider Using condition (30), , for and , we have
Lemma 9. Let be a nonnegative and nonincreasing sequence satisfies (30); then , we get
Proof. Consider Using condition (30), for all and , we have
Lemma 10 (see [19]). If satisfies conditions (28) and (29), then
5. Proof of Theorem 5
The integral representationof is given by and, therefore, Denoting means of by , we have Now, using condition (30), then transform of transform is given by Set Now, using (34) and Lemma 10, assume that satisfies (28) and (29); we get Now, using (34) and Lemma 10, assume that satisfies (28) and (29); we get Now, using (33), Lemma 8, we get Now, using (33), Lemma 9, we get Using the fact that we may write , and combining (47) and (49), we get Combining (48) and (50), we get Therefore, from (8), (51), and (52), we have Since it follows from (47) and (48) that Combining (53) and (55), we get To prove (32), if satisfies only (28), then, using (34) and second mean value theorem for integrals, we have Combining (49) and (57), we get Again, if satisfies only (28), then using (34) and second mean value theorem for integrals, we have Combining (50) and (59), we get Therefore, Combining (61) and (62), we get This completes the proof of Theorem 5.
6. Applications
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. As mentioned in [21], the space in general and and in particular play an important role in the theory of signals and filters. From the point of view of the applications, sharper estimates of infinite matrices [25] are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix valued functions and enable to investigate perturbations of matrix valued functions and compare them. The following corollaries may be derived from Theorem 5.
If and set then we get Corollary 11.
Corollary 11. If , then If we put in Corollary 11, then we get Corollary 12.
Corollary 12. If Lip , then
7. Example
In the example, we see how the sequence of averages (i.e., -means or mean) and Nörlund mean of partial sums of a Fourier series is better behaved than the sequence of partial sums itself.
Let with for all real . Fourier series of is given by Then th partial sum of Fourier series (68) and th Cesàro sum for , that is, for the series (68), are given by From Theorem 20 of Hardy’s “Divergent Series,” if a Nörlund method has increasing weights , then it is stronger than .
Now, take to be the Nörlund matrix generated by , then Nörlund means is given by In this graph, we observe that converges to faster than in the interval . We further note that near the points of discontinuities, that is, , and , the graph of and shows peaks that move closer to the line passing through points of discontinuity as n increases (Gibbs phenomenon), but in the graph of the peaks become flatter (Figure 1). The Gibbs phenomenon is an overshoot a peculiarity of the Fourier series and other eigen function series at a simple discontinuity; that is, the convergence of Fourier series is very slow near the point of discontinuity. Thus, the product summability means of the Fourier series of overshoot the Gibbs Phenomenon and show the smoothing effect of the method. Thus, is the better approximant than and method is stronger than () method.
(a)
(b)
8. Conclusion
Several results concerning the degree of approximation of periodic signals (functions) by product summability means of Fourier series and conjugate Fourier series in generalized Hölder metric and Hölder metric have been reviewed. Using graphical representation is a better approximant to , but till now nothing has been done to show this. Some interesting application of the operator used in this paper is pointed out in Remark 6. Also, the result of our theorem is more general rather than the results of any other previous proved theorems, which will enrich the literate of summability theory of infinite series.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this paper. Special thanks are due to Professor Ricardo Estrada, Academic Editor of IJMMS, and Professor Mohammed Mahmoud, Editorial office, Hindawi Publishing Corporation, for kind cooperation, smooth behavior during communication, and their efforts to send the reports of the paper timely. The authors are also grateful to all the editorial board members and reviewers of Hindawi prestigious journal, that is, International Journal of Mathematics and Mathematical Sciences. The second author Kejal Khatri acknowledges the Ministry of Human Resource Development, New Delhi, India, for providing the financial support and to carry out her research work under the guidance of Dr. Vishnu Narayan Mishra at S.V.N.I.T., Surat (Gujarat), India. The first author Vishnu Narayan Mishra acknowledges Cumulative Professional Development Allowance (CPDA), SVNIT, Surat (Gujarat), India, for supporting this research paper. Both authors carried out the proof. All the authors conceived of the study and participated in its design and coordination. Both the authors read and approved the final manuscript.