Error Bound of Periodic Signals in the Hölder Metric

Abstract

We obtain two theorems to determine the error bound between input periodic signals and processed output signals, whenever signals belong to [subscript]Hω[/subscript] -space and as a processor we have taken (C,1)(E,1) -mean and generalized an early result of Lal and Yadav in (2001).

Tikam Singh 1 and Pravin Mahajan 2

Recommended by Narendra Kumar Govil

1, 121 Mahashweta Nagar, Ujjain 456010, India
2, Department of Mathematics, Jawaharlal Institute of Technology, Borawan 451228, Dist. Khargone, India

Received 16 October 2007; Revised 16 January 2008; Accepted 9 March 2008

1. Introduction

Chandra [1] was first to extend Prössdorf's [2] result to find the degree of approximation of a continuous function using the Nörlund transform. Later on, Mohapatra and Chandra [3] obtained a number of interesting results on the degree of approximation in the Hölder metric using matrix transforms, which generalize all the previous results based on Cesàro and Nörlund transforms. In 1992, Singh [4] introduced _Hω -space in place of _Hα -space and obtained several results on the degree of approximation of functions and deduced many previous results based on _Hα -spaces. In 1996, Das et al. [5] used _H(α,p) -space in place of _Hα -space and obtained degree of approximation of functions and generalized the results of Mohapatra and Chandra [3]. In 2000, Mittal and Rhoades [6] also obtained the degree of approximation of functions in a normed space and generalized the results of Singh [4] by removing the hypothesis of monotonicity of the rows of the matrix. Singh and Soni [7], and Mittal et al. [8] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals.

2. Definitions and Notations

Let the transforms

A:: [figure omitted; refer to PDF]

B:: [figure omitted; refer to PDF]

be two regular methods of summability. Then, the A transform of the B transform of a sequence {_sn } is given by [figure omitted; refer to PDF] the sequence {_sn } is said to be summable _tn to the sum s , if [figure omitted; refer to PDF] Let s(t)∈_C2π be a 2π -periodic analog signal whose Fourier trigonometric expansion be given by [figure omitted; refer to PDF] and let {_sn (t)} be the sequence of partial sums of (2.5).

Let the (E,1) and (C,1) transforms for the sequence {_sn } be defined by [figure omitted; refer to PDF] [figure omitted; refer to PDF] respectively.

The product (C,1)(E,1) -transform is expressed as the (C,1) -transform of (E,1) -transform of {_sn } and is given by sequence-to-sequence transformation (see, e.g., [9]): [figure omitted; refer to PDF] The sequence {_sn } is said to be summable (C,1)(E,1) to the sum s , if [figure omitted; refer to PDF]

2.1. Regularity Condition of (C,1)(E,1) -Method

[figure omitted; refer to PDF] where [figure omitted; refer to PDF] Now,

(i) ^∑k=0∞ |_Cn,k |=^∑k=0n |(1/(n+1))^{2-k∑[upsilon]=0k} (k[upsilon])|=1,

(ii) _Cn,k =(1/(n+1))(1)[arrow right]0, as n[arrow right]∞ , for fixed k ,

(iii): ^∑k=0∞ _Cn,k =1,

thus, (C,1)(E,1) -method is regular.

Singh [4] defined the space _Hω by

[figure omitted; refer to PDF] and the norm _{||[sm middot]||^{ω[low *]}} by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and choosing ^Δ0 s(_t1 ,_t2 )=0 , ω(t) and ^{ω[low *]} (t) being increasing signals of t . If ω(|_t1 -_t2 |)≤A^{|_t1 -_t2 |α} and ^{ω[low *]} (|_t1 -_t2 |)≤K^{|_t1 -_t2 |β} , 0≤β<α≤1 , A and K being positive constants, then the space [figure omitted; refer to PDF] is Banach space [2] and the metric induced by the norm _{||[sm middot]||α} on _Hα is said to be Hölder metric.

We write [figure omitted; refer to PDF] [figure omitted; refer to PDF]

3. Known Result

Lal and Yadav [10] established the following theorem to estimate the error between the input signal s(t) and the signal obtained after passing through the (C,1)(E,1) -transform.

Theorem A.

If a function s:R[arrow right]R is 2π -periodic and belonging to class Lip α , 0<α≤1 , then the degree of approximation by (C,1)(E,1) means of its Fourier series is given by

[figure omitted; refer to PDF]

4. Main Result

The object of this paper is to generalize the above result under much more general assumptions. We will measure the error between the input signal s(t) and the processed output signal _tn (s;t)=(1/(n+1))^{∑k=1n Ek1} (t) , by establishing the following theorems.

Theorem 4.1.

Let ω(t) defined in (2.12) be such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] then, for 0≤β<η≤1 and s∈_Hω , we have [figure omitted; refer to PDF]

Theorem 4.2.

Let ω(t) defined in (2.12) and for 0≤β<η≤1 and s∈_Hω , we have [figure omitted; refer to PDF]

5. Lemmas

We will use following lemmas.

Lemma 5.1.

Let _[varphi]t1 (t) be defined in (2.16), then for s∈_Hω , we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] It is easy to verify.

Lemma 5.2.

Let _Kn (t) be defined in (2.17), then [figure omitted; refer to PDF] where " C " is an absolute constant, not necessarily the same at each occurrence.

Proof.

[figure omitted; refer to PDF]

Lemma 5.3.

[figure omitted; refer to PDF]

Proof.

[figure omitted; refer to PDF]

Lemma 5.4 (see [9]).

For 0≤t≤1/n+1 , then [figure omitted; refer to PDF]

Lemma 5.5 (see [6]).

If ω(t) satisfies conditions (4.1) and (4.2), then [figure omitted; refer to PDF]

6. Proof of Theorem 4.1

Proof of Theorem 4.1.

Following Zygmund [11], we have [figure omitted; refer to PDF] From (2.6) and (2.16), we have [figure omitted; refer to PDF] Using Lemma 5.2, we have [figure omitted; refer to PDF] Now from (2.8), the (C,1) -transform of (E,1) -transform is given by [figure omitted; refer to PDF] Setting [figure omitted; refer to PDF] now using (4.1), (4.2), (5.2), and Lemma 5.5, we get [figure omitted; refer to PDF] Again using (5.2), (4.1), and Lemma 5.3, we have [figure omitted; refer to PDF] Now from (5.1), Lemmas 5.3 and 5.4, we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] Now noting that [figure omitted; refer to PDF] we have, from (6.6) and (6.8), [figure omitted; refer to PDF] and from (6.7) and (6.9), we have [figure omitted; refer to PDF] Thus, from (2.13), (6.11) and (6.12), we have [figure omitted; refer to PDF] It is to be noted from (6.6) and (6.7), [figure omitted; refer to PDF] Combining (6.13) and (6.14), we get [figure omitted; refer to PDF] This completes the proof of Theorem 4.1.

Proof of Theorem 4.2.

Follows analogously as the proof of Theorem 4.1 with slight changes, so we omit details.

7. Applications

The following results can easily be derived from the Theorem 4.1. If we put ^{ω[low *]} (|_t1 -_t2 |)≤K^{|_t1 -_t2 |β} , ω(|_t1 -_t2 |)≤A^{|_t1 -_t2 |α} and replace η by α and set [figure omitted; refer to PDF] then we get Corollary 7.1.

Corollary 7.1.

If s∈_Hα , 0≤β<α≤1 , then [figure omitted; refer to PDF] If we put β=0 , then from above corollary, we have Corollary 7.2.

Corollary 7.2.

If s∈Lip α , 0<α≤1 , then [figure omitted; refer to PDF] Hence Theorem 3 is particular case of Theorem 4.1.

Acknowledgment

The authors are highly grateful to the referee for his valuable comments and suggestions for the improvement and the better presentation of the paper.

References

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[2] S. Prössdorf, "Zur Konvergenz der Fourierreihen Hölder stetiger Funktionen," Mathematische Nachrichten , vol. 69, no. 1, pp. 7-14, 1975.

[3] R. N. Mohapatra, P. Chandra, "Degree of approximation of functions in the Hölder metric," Acta Mathematica Hungarica , vol. 41, no. 1-2, pp. 67-76, 1983.

[4] T. Singh, "Degree of approximation to functions in a normed space," Publicationes Mathematicae Debrecen , vol. 40, no. 3-4, pp. 261-271, 1992.

[5] G. Das, T. Ghosh, B. K. Ray, "Degree of approximation of functions by their Fourier series in the generalized Hölder metric," Proceedings of the Indian Academy of Sciences. Mathematical Sciences , vol. 106, no. 2, pp. 139-153, 1996.

[6] M. L. Mittal, B. E. Rhoades, "Degree of approximation to functions in a normed space," Journal of Computational Analysis and Applications , vol. 2, no. 1, pp. 1-10, 2000.

[7] T. Singh, B. Soni, "Approximation by generalized de la Vallee Poussin operators," The Mathematics Student , vol. 74, no. 1-4, pp. 199-206, 2005.

[8] M. L. Mittal, B. E. Rhoades, V. N. Mishra, "Approximation of signals (functions) belonging to the weighted W(_Lp ,ξ(t)) -class by linear operators," International Journal of Mathematics and Mathematical Sciences , vol. 2006, 2006.

[9] S. Lal, P. N. Singh, "On approximation of Lip (ξ(t),p) function by (C,1)(E,1) means of its Fourier series," Indian Journal of Pure and Applied Mathematics , vol. 33, no. 9, pp. 1443-1449, 2002.

[10] S. Lal, K. N. Singh Yadav, "On degree of approximation of function belonging to the Lipschitz class by (C,1)(E,1) means of its Fourier series," Bulletin of the Calcutta Mathematical Society , vol. 93, no. 3, pp. 191-196, 2001.

[11] A. Zygmund Trigonometric Series , vol. 1, pp. xii+383 pp, Cambridge University Press, New York, NY, USA, 1959.