Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 495075, 9 pages
doi:10.1155/2008/495075
Research Article
Error Bound of Periodic Signals in
the Hölder Metric
Tikam Singh1 and Pravin Mahajan2
1
2
121 Mahashweta Nagar, Ujjain 456010, India
Department of Mathematics, Jawaharlal Institute of Technology, Borawan 451228,
Dist. Khargone, India
Correspondence should be addressed to Pravin Mahajan, pravin mahajan75@yahoo.com
Received 16 October 2007; Revised 16 January 2008; Accepted 9 March 2008
Recommended by Narendra Kumar Govil
We obtain two theorems to determine the error bound between input periodic signals and processed
output signals, whenever signals belong to Hω -space and as a processor we have taken C, 1E, 1mean and generalized an early result of Lal and Yadav in 2001.
Copyright q 2008 T. Singh and P. Mahajan. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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 Hα,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.
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International Journal of Mathematics and Mathematical Sciences
2. Definitions and notations
Let the transforms
A:
λn n
ank sk ,
2.1
bnk sk ,
2.2
k1
B:
τn n
k1
be two regular methods of summability. Then, the A transform of the B transform of a sequence
{sn } is given by
tn n
anp τp p1
n
n anp bpk sk ,
2.3
p1 k1
the sequence {sn } is said to be summable tn to the sum s, if
Lim tn s.
n→∞
2.4
Let st ∈ C2π be a 2π-periodic analog signal whose Fourier trigonometric expansion be given
by
∞
∞
1
st ∼ a0 An t,
an cos nt bn sin nt ≡
2
n0
n1
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
n
1
n
1
En n
sk t,
2 k0 k
σn n
1 sk t,
n 1 k0
2.5
2.6
2.7
respectively.
The product C, 1E, 1-transform is expressed as the C, 1-transform of E, 1transform of {sn } and is given by sequence-to-sequence transformation see, e.g., 9:
tn s; t n
1 E1 .
n 1 k0 k
2.8
The sequence {sn } is said to be summable C, 1E, 1 to the sum s, if
Lim tn s; t s.
n→∞
2.9
T. Singh and P. Mahajan
3
2.1. Regularity condition of C, 1E, 1-method
k
n
n
∞
1 1 1
k
1
tn s; t Ek Cn,k sk ,
s
k
n 1 k0
n 1 k0 2k υ0 υ
k0
where
⎧
k
⎪
k
⎪
⎨ 1 2−k
, k≤n
n1
υ
υ0
⎪
⎪
⎩0,
k > n.
Cn,k
2.10
2.11
Now,
i
∞
k0
|Cn,k | n
k0 |1/n
12−k
k
k
υ0 υ |
1,
ii Cn,k 1/n 11 → 0, as n → ∞, for fixed k,
iii ∞
k0 Cn,k 1,
thus, C, 1E, 1-method is regular.
Singh 4 defined the space Hω by
Hω st ∈ C2π : s t1 − s t2 ≤ Kω t1 − t2 ,
2.12
and the norm · ω∗ by
∗ sω∗ sc Sup Δω s t1 , t2 ,
2.13
t1 ,t2
where
sc Sup |st|,
0≤t≤2π
ω∗
Δ s t1 , t2
s t1 − s t2 ,
ω∗ t1 − t2 2.14
t1 /
t2 ,
and choosing Δ0 st1 , t2 0, ωt and ω∗ t being increasing signals of t. If ω|t1 − t2 | ≤
A|t1 − t2 |α and ω∗ |t1 − t2 | ≤ K|t1 − t2 |β , 0 ≤ β < α ≤ 1, A and K being positive constants,
then the space
α
Hα st ∈ C2π : s t1 − s t2 ≤ K t1 − t2 , 0 < α ≤ 1
2.15
is Banach space 2 and the metric induced by the norm · α on Hα is said to be Hölder metric.
We write
φt1 t s t1 t s t1 − t − 2s t1 ,
n
t
1
n
t.
sin k Kn t sinn 1
2 k0 k
2
2.16
2.17
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International Journal of Mathematics and Mathematical Sciences
3. Known result
Lal and Yadav 10 established the following theorem to estimate the error between the input
signal st and the signal obtained after passing through the C, 1E, 1-transform.
Theorem A. If a function s : R → R is 2π-periodic and belonging to class Lip α, 0 < α ≤ 1, then the
degree of approximation by C, 1E, 1 means of its Fourier series is given by
⎧ −α
⎪
0<α<1
⎨O n ,
tn s; t1 − s t1 log n
∞
⎪
⎩O
, α 1.
n
3.1
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 st and the processed output signal
tn s; t 1/n 1 nk1 Ek1 t, by establishing the following theorems.
Theorem 4.1. Let ωt defined in 2.12 be such that
π
t
t
ωu
du O Ht ,
2
u
Hudu O tHt ,
4.1
Ht ≥ 0,
as t −→ 0 ,
4.2
0
then, for 0 ≤ β < η ≤ 1 and s ∈ Hω , we have
tn s; t1 − s
ω∗
O
π
n 1 H
n1
−1
1−β/η .
4.3
Theorem 4.2. Let ωt defined in 2.12 and for 0 ≤ β < η ≤ 1 and s ∈ Hω , we have
tn s; t1 − s ∗ O
ω
1−β/η 1−β/η n1 π
1
−1
ω
n 1
ω
.
1
n
k1
k1
4.4
5. Lemmas
We will use following lemmas.
Lemma 5.1. Let φt1 t be defined in 2.16, then for s ∈ Hω , we have
φt t − φt t ≤ 4Kω t1 − t2 ,
1
2
φt t − φt t ≤ 4Kω |t| .
1
2
It is easy to verify.
5.1
5.2
T. Singh and P. Mahajan
5
Lemma 5.2. Let Kn t be defined in 2.17, then
n1 2
t
n t
Kn t ≤ C
cos
sinn 1
,
t
2
2
5.3
where “C” is an absolute constant, not necessarily the same at each occurrence.
Proof.
n
1
n
eik1/2t
I.P.
Kn t k
sint/2
k0
n
1
I.P. eit/2 1 eit
sint/2
t in1t/2
1
e
I.P. 2n cosn
sint/2
2
n1 2
t
t
≤C
cosn
sinn 1
.
t
2
n
5.4
n t
C
1
t
t
t
cosk
sink 1
≤ 2
cosn1
.
1 − cosn 1
t
2
2
2
2
t
k0
5.5
Lemma 5.3.
Proof.
n t
1
t
cosk
sink 1
t
2
2
k0
n 1
t
I.P. eik1t/2 cosk
t
2
k0
1 − ein1t/2 cosn1 t/2
1
I.P. eit/2
t
1 − eit/2 cost/2
C
t
t
t
t
≤ 2 I.P. i − i cosn 1
cosn1
sinn 1
cosn1
2
2
2
2
t
t
t
C
cosn1
.
1 − cosn 1
2
2
2
t
Lemma 5.4 see 9. For 0 ≤ t ≤ 1/n 1, then
t
n1 t
cos
O n 12 t2 .
1 − cosn 1
2
2
Lemma 5.5 see 6. If ωt satisfies conditions 4.1 and 4.2, then
u
t−1 ωtdt O uHu , u −→ 0 .
0
5.6
5.7
5.8
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International Journal of Mathematics and Mathematical Sciences
6. Proof of Theorem 4.1
Proof of Theorem 4.1. Following Zygmund 11, we have
1
sn t1 − s 2π
π
0
φt1 t
1
sin n t dt.
sint/2
2
6.1
From 2.6 and 2.16, we have
En1
2−n
t1 − s 2π
π
φt1 tKn tdt.
6.2
0
Using Lemma 5.2, we have
En1
2−n1
t1 − s ≤ C
π
π
0
φt1 t n1 n t
t
sinn 1
dt.
2 cos
t
2
2
6.3
Now from 2.8, the C, 1-transform of E, 1-transform is given by
tn s; t1 − s ≤
C
n1
n
π φt1 t t k t
sink 1
cos
dt.
t
2
2
0
k0
6.4
Setting
n
π φt1 t t k t
sink 1
cos
dt,
t
2
2 0
k0
n
π φt1 t − φt2 t t t
C
k
sink 1
En t1 , t2 En t1 − En t2 ≤
cos
dt
k0
n1 0
t
2
2 En t1 tn s; t1 − s t1 ≤
O
1
n1
C
n1
π/n1 π
0
I1 I2 ,
say,
π/n1
6.5
now using 4.1, 4.2, 5.2, and Lemma 5.5, we get
1
I1 O1
n1
π/n1
0
π
.
t−1 ωtdt O n 1−1 H
n1
6.6
Again using 5.2, 4.1, and Lemma 5.3, we have
I2 O1
1
n1
O1
1
n1
π
t
t dt
cosn1
t−2 ωt1 − cosn 1
2
2 π/n1
π
π/n1
t−2 ωtdt
π
.
O n 1−1 H
n1
6.7
T. Singh and P. Mahajan
7
Now from 5.1, Lemmas 5.3 and 5.4, we have
π/n1 ω t1 − t2 1
1 − cosn 1 t cosn1 t dt
I1 O1
2
n1 0
2
2 t
π/n1
ω t1 − t2 O1
t−2 n 12 t2 dt
n1
0
O ω t1 − t2 ,
ω t1 − t2 π
I2 O1
t−2 dt
n1
π/n1
O ω t1 − t2 .
6.8
6.9
Now noting that
1−β/η β/η
Ir ,
I r Ir
r 1, 2,
6.10
we have, from 6.6 and 6.8,
1−β/η β/η
π
−1
I1 O ω t1 − t2
,
n 1 H
n1
and from 6.7 and 6.9, we have
1−β/η β/η
π
−1
n 1 H
I2 O ω t1 − t2
.
n1
6.11
6.12
Thus, from 2.13, 6.11 and 6.12, we have
En t1 − En t2 ω∗ supΔ En t1 , t2 sup
ω∗ |t1 − t2 |
t1 ,t2
t1 ,t2
1−β/η β/η ∗ −1
π
−1
n 1 H
O ω t1 − t2
.
ω t1 − t2
n1
6.13
It is to be noted from 6.6 and 6.7,
π
En t1 max tn s; t1 − s O n 1−1 H
.
c
0≤t1 ≤2π
n1
6.14
Combining 6.13 and 6.14, we get
tn s; t1 − s
ω∗
O
π
n 1 H
n1
−1
1−β/η .
6.15
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.
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International Journal of Mathematics and Mathematical Sciences
7. Applications
The following results can easily be derived from the Theorem 4.1. If we put ω∗ |t1 − t2 | ≤
K|t1 − t2 |β , ω|t1 − t2 | ≤ A|t1 − t2 |α and replace η by α and set
⎧
α−1
⎪
0<α<1
⎨u ,
Hu 1
⎪
⎩log
, α 1,
u
7.1
then we get Corollary 7.1.
Corollary 7.1. If s ∈ Hα , 0 ≤ β < α ≤ 1, then
⎧
β−α
⎪
0<α<1
⎪
⎨On 1 ,
tn s; t1 − s 1−β
β
logn 1
⎪
⎪
, α 1.
⎩O
n 1
7.2
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
⎧ −α
⎪
0<α<1
⎨O n ,
tn s; t1 − s log n
⎪
⎩O
, α 1.
n
7.3
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.
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