A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging to -class by product summability means of conjugate series of Fourier series has been generalized for the weighted , -class, where is nonnegative and increasing function of , by which is in more general form of Theorem 2 of Nigam and Sharma (2011).
1. Introduction
Khan [1, 2] has studied the degree of approximation of a function belonging to and classes by NΓΆrlund and generalized NΓΆrlund means. Working in the same direction Rhoades [3], Mittal et al. [4], Mittal and Mishra [5], and Mishra [6, 7] have studied the degree of approximation of a function belonging to class by linear operators. Thereafter, Nigam [8] and Nigam and Sharma [9] discussed the degree of approximation of conjugate of a function belonging to class and by product summability means, respectively. Recently, Rhoades et al. [10] have determined very interesting result on the degree of approximation of a function belonging to class by Hausdorff means. Summability techniques were also applied on some engineering problems like Chen and Jeng [11] who implemented the CesΓ ro 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 et al. [12] applied regularization with CesΓ ro sum technique for the derivative of the double layer potential. Similarly, Chen and Hong [13] used CesΓ ro sum regularization technique for hypersingularity of dual integral equation.
The generalized weighted -class is generalization of and classes. Therefore, in the present paper, a theorem on degree of approximation of conjugate of signals belonging to the generalized weighted class by product summability means of conjugate series of Fourier series has been established which is in more general form than that of Nigam and Sharma [9]. We also note some errors appearing in the paper of Nigam [8], Nigam and Sharma [9] and rectify the errors pointed out in Remarks 2.2, 2.3 and 2.4.
Let be a -periodic function and integrable in the sense of Lebesgue. The Fourier series of at any point is given by
with th partial sum .
The conjugate series of Fourier series (1.1) is given by
Let be a given infinite series with sequence of its th partial sums . The transform is defined as the th partial sum of summability, and we denote it by .
If
then the infinite series is summable to a definite number , Hardy [14].
If
then the infinite series is summable to the definite number by method. The transform of the transform defines product transform and denotes it by . Thus, if
then the infinite series is said to be summable by method or summable to a definite number . The is regular method of summability
A function , if
and , for , if
Given a positive increasing function , if
Given positive increasing function , an integer , ([2]), if
For our convenience to evaluate without error, we redefine the weighted class as follows:
If , then the weighted class coincides with the class , we observe that
-norm of a function is defined by
A signal is approximated by trigonometric polynomials of order , and the degree of approximation is given by Rhoades [3]
in terms of , where is a trigonometric polynomial of degree . This method of approximation is called trigonometric Fourier approximation (TFA) [4].
We use the following notations throughout this paper:
2. Previous Result
Nigam [8] has proved a theorem on the degree of approximation of a function , conjugate to a periodic function with period and belonging to the class by product summability means of conjugate series of Fourier series. He has proved the following theorem.
Theorem 2.1 (see [8]). If , conjugate to a -periodic function , belongs 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 is an arbitrary number such that , condition (2.2) and (2.3) hold uniformly in and is means of the series (1.2).
Remark 2.2. The proof proceeds by estimating , which is represented in terms of an integral. The domain of integration is divided into two partsβfrom and . Referring to second integral as , and using HΓΆlder inequality, the author [8] obtains
The author then makes the substitution to obtain
In the next step is removed from the integrand by replacing it with , while is an increasing function, is now a decreasing function. Therefore, this step is invalid.
Remark 2.3. The proof follows by obtaining , in Theorem 2 of Nigam and Sharma [9], which is expressed in terms of an integral. The domain of integration is divided into two partsβfrom and . Referring to second integral as , and using HΓΆlder inequality, the authors [9] obtain the following:
The authors then make the substitution to get
In the next step, is removed from the integrand by replacing it with , while is an increasing function, is now a decreasing function. Therefore, in view of second mean value theorem of integral, this step is invalid.
Remark 2.4. The condition used by Nigam and Sharma [9] is not valid, since as .
3. Main Result
It is well known that the theory of approximation, that is, TFA, which originated from a 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 [15], in general, and in digital signal processing [16] in particular, in view of the classical Shannon sampling theorem. Broadly speaking, signals are treated as function of one variable and images are represented by functions of two variables.
This has motivated Mittal and Rhoades [17β20] and Mittal et al. [4, 21] to obtain many results on TFA using summability methods without rows of the matrix. In this paper, we prove the following theorem.
Theorem 3.1. If , conjugate to a -periodic function , belongs to the generalized weighted -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 is an arbitrary number such that , conditions (3.2) and (3.3) hold uniformly in and is means of the series (1.2) and the conjugate function is defined for almost every by
Note 1. , for .
Note 2. Also for , Theorem 3.1 reduces to Theorem 2.1, and thus generalizes the theorem of Nigam [8]. Also our Theorem 3.1 in the modified form of Theorem 2 of Nigam and Sharma [9].
Note 3. The product transform plays an important role in signal theory as a double digital filter [6] and the theory of machines in mechanical engineering.
4. Lemmas
For the proof of our theorem, the following lemmas are required.
Lemma 4.1. Consider for .
Proof. For and .
This completes the proof of Lemma 4.1.
Lemma 4.2. Consider , for and any .
Proof. For .
Now considering first term of (4.2)
Now considering second term of (4.2) and using Abelβs lemma
On combining (4.2), (4.3), and (4.4)
This completes the proof of Lemma 4.2.
5. Proof of Theorem
Let denotes the th partial sum of series (1.2). Then following Nigam [8], we have
Therefore, using (1.2), the transform of is given by
Now denoting transform of as , we write
We consider, .
Using HΓΆlderβs inequality
Since is a positive increasing function and by using second mean value theorem for integrals, we have
Note that ,
Now, we consider
Using HΓΆlderβs inequality
Now putting , we have
Since is a positive increasing function, so is also a positive increasing function and using 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 3.1.
6. Applications
The theory of approximation is a very extensive field, which has various applications, and the study of the theory of trigonometric Fourier approximation is of great mathematical interest and of great practical importance. From the point of view of the applications, Sharper estimates of infinite matrices [22] are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix valued functions and enables to investigate perturbations of matrix valued functions and compare them.
The following corollaries may be derived from Theorem 3.1.
Corollary 6.1. If , then the weighted class reduces to the class and the degree of approximation of a function conjugate to a -periodic function belonging to the class , is given by
Proof. The result follows by setting in (3.1), we have
Thus, we get
This completes the proof of Corollary 6.1.
Corollary 6.2. If for and in Corollary 6.1, then and
Proof. For in Corollary 6.1, we get
Thus, we get
This completes the proof of Corollary 6.2.
Acknowledgments
The authors are highly thankful to the anonymous referees for the careful reading, their critical remarks, valuable comments, and several useful suggestions which helped greatly for the overall improvements and the better presentation of this paper. The authors are also grateful to all the members of editorial board of IJMMS, especially Professor Noran El-Zoheary, Editorial Staff H. P. C. and Professor Ram U. Verma, Texas A&M University, USA, the Editor of IJMMS, Omnia Kamel Editorial office, HPC, Riham Taha Accounts Receivable specialist, HPC for their kind cooperation during communication. The authors are also thankful to the Cumulative Professional Development Allowance (CPDA) SVNIT, Surat (Gujarat), India, for their financial assistance.
References
H. H. Khan, βOn the degree of approximation of functions belonging to class Lip(α,p),β Indian Journal of Pure and Applied Mathematics, vol. 5, no. 2, pp. 132β136, 1974.
H. H. Khan, βOn the degree of approximation to a function belonging to weighted -class,β The Aligarh Bulletin of Mathematics, vol. 3-4, pp. 83β88, 1973/74.
B. E. Rhoades, βOn the degree of approximation of the conjugate of a function belonging to the weighted class by matrix means of the conjugate series of a Fourier series,β Tamkang Journal of Mathematics, vol. 33, no. 4, pp. 365β370, 2002.
M. L. Mittal, B. E. Rhoades, and V. N. Mishra, βApproximation of signals (functions) belonging to the weighted -class by linear operators,β International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 53538, 10 pages, 2006.
M. L. Mittal and V. N. Mishra, βApproximation of signals (functions) belonging to the Weighted -Class by almost matrix summability method of its Fourier series,β International Journal of Mathematical Sciences and Engineering Applications, vol. 2, pp. 1β9, 2008.
V. N. Mishra, βOn the degree of approximation of signals (functions) belonging to Generalized Weighted -class by product summability method,β Journal of International Academy of Physical Sciences, vol. 14, no. 4, pp. 413β423, 2010.
V. N. Mishra, βOn the Degree of Approximation of signals (functions) belonging to Generalized Weighted -Class by almost matrix summability method of its conjugate Fourier series,β International Journal of Applied Mathematics and Mechanics, vol. 5, pp. 16β27, 2009.
H. K. Nigam, βApproximation of a conjugate of a function -class by product means of conjugate series of Fourier series,β Ultra Scientist, vol. 22, no. 1, pp. 295β302, 2010.
H. K. Nigam and K. Sharma, βApproximation of conjugate of functions belonging to class and class by product means of conjugate fourier series,β European Journal of Pure and Applied Mathematics, vol. 4, no. 3, pp. 276β286, 2011.
B. E. Rhoades, K. Ozkoklu, and I. Albayrak, βOn the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its Fourier series,β Applied Mathematics and Computation, vol. 217, no. 16, pp. 6868β6871, 2011.
J. T. Chen and Y. S. Jeng, βDual series representation and its applications to a string subjected to support motions,β Advances in Engineering Software, vol. 27, pp. 227β238, 1996.
J. T. Chen, H.-K. Hong, C. S. Yeh, and S. W. Chyuan, βIntegral representations and regularizations for a divergent series solution of a beam subjected to support motions,β Earthquake Engineering and Structural Dynamics, vol. 25, pp. 909β925, 1996.
J. T. Chen and H.-K. Hong, βReview of dual boundary element methods with emphasis on hypersingular integrals and divergent series,β ASME Applied Mechanics Reviews, vol. 52, pp. 17β33, 1999.
J. G. Proakis, Digital Communications, McGraw-Hill, New York, NY, USA, 1985.
E. Z. Psarakis and G. V. Moustakides, βAn -based method for the design of 1-D zero phase FIR digital filters,β IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 44, no. 7, pp. 591β601, 1997.
M. L. Mittal and B. E. Rhoades, βOn the degree of approximation of continuous functions by using linear operators on their Fourier series,β International Journal of Mathematics, Game Theory, and Algebra, vol. 9, no. 4, pp. 259β267, 1999.
M. L. Mittal and B. E. Rhoades, βDegree of approximation of functions in the Hölder metric,β Radovi Matematički, vol. 10, no. 1, pp. 61β75, 2001.
M. L. Mittal and B. E. Rhoades, βDegree of approximation to functions in a normed space,β Journal of Computational Analysis and Applications, vol. 2, no. 1, pp. 1β10, 2000.
M. L. Mittal and B. E. Rhoades, βApproximation by matrix means of double Fourier series to continuous functions in two variables,β Radovi Matematički, vol. 9, no. 1, pp. 77β99, 1999.
M. L. Mittal, B. E. Rhoades, V. N. Mishra, S. Priti, and S. S. Mittal, βApproximation of functions (signals) belonging to -class by means of conjugate Fourier series using linear operators,β Indian Journal of Mathematics, vol. 47, no. 2-3, pp. 217β229, 2005.
M. L. Mittal, B. E. Rhoades, S. Sonker, and U. Singh, βApproximation of signals of class by linear operators,β Applied Mathematics and Computation, vol. 217, no. 9, pp. 4483β4489, 2011.