# Error Bound of Periodic Signals in  the Hölder Metric

Authors: Tikam Singh; Pravin Mahajan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

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


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## Body

## 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 transformsA:
(2.1)λn=∑k=1nanksk,B:
(2.2)τn=∑k=1nbnksk,
be
two regular methods of summability. Then, the A transform of the B transform
of a sequence {sn} is given by (2.3)tn=∑p=1nanpτp=∑p=1n∑k=1nanpbpksk,the
sequence {sn} is said to be summable tn to the sum s, if (2.4)Limn→∞tn=s. Let s(t)∈C2π be a 2π-periodic analog signal whose Fourier
trigonometric expansion be given by (2.5)s(t)∼12a0+∑n=1∞(ancosnt+bnsinnt)≡∑n=0∞An(t), 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 (2.6)En1=12n∑k=0n(nk)sk(t),(2.7)σn=1n+1∑k=0nsk(t), 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]): (2.8)tn(s;t)=1n+1∑k=0nEk1. The sequence {sn} is said to be summable (C,1)(E,1) to the sum
s, if (2.9)Limn→∞tn(s;t)=s.


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

(2.10)tn(s;t)=1n+1∑k=0nEk1=1n+1∑k=0n{12k∑υ=0k(kυ)sk}=∑k=0∞Cn,ksk, where(2.11)Cn,k={1n+12−k∑υ=0k(kυ),k≤n0,k>n.Now,(i)
∑k=0∞|Cn,k|=∑k=0n|(1/(n+1))2−k∑υ=0k(kυ)|=1,(ii)
Cn,k=(1/(n+1))(1)→0, as n→∞,
for fixed k,(iii)
∑k=0∞Cn,k=1,
thus, (C,1)(E,1)-method is regular.Singh [4] defined the space Hω by(2.12)Hω={s(t)∈C2π:|s(t1)−s(t2)|≤Kω(|t1−t2|)}, and
the norm ∥⋅∥ω∗ by (2.13)∥s∥ω∗=∥s∥c+Supt1,t2{Δω∗s(t1,t2)}, where (2.14)∥s∥c=Sup0≤t≤2π|s(t)|,Δω∗s(t1,t2)=|s(t1)−s(t2)|ω∗(|t1−t2|),t1≠t2,and
choosing Δ0s(t1,t2)=0, ω(t) 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 (2.15)Hα={s(t)∈C2π:|s(t1)−s(t2)|≤K|t1−t2|α,0<α≤1} is Banach
space [2] and the metric induced by the norm ∥⋅∥α on Hα is said to be Hölder metric.We write(2.16)ϕt1(t)=s(t1+t)+s(t1−t)−2s(t1),(2.17)Kn(t)=sin(n+1)t2∑k=0n(nk)sin(k+12)t.


### 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 functions:R→R is 2π-periodic and
belonging to class Lipα, 0<α≤1,
then the degree of approximation by
(C,1)(E,1) means of its Fourier series is given by(3.1)∥tn(s;t1)−s(t1)∥∞={O(n−α),0<α<1O(lognn),α=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 signals(t) and the processed output
signal tn(s;t)=(1/(n+1))∑k=1nEk1(t),
by establishing the following theorems.Theorem 4.1.
Letω(t) defined in (2.12) be such that (4.1)∫tπω(u)u2du=O{H(t)},H(t)≥0,(4.2)∫0tH(u)du=O{tH(t)},ast→0+, then, for 0≤β<η≤1 and s∈Hω,
we have (4.3)∥tn(s;t1)−s∥ω∗=O{((n+1)−1H(πn+1))1−β/η}.Theorem 4.2.
Letω(t) defined in (2.12) and for 0≤β<η≤1 and s∈Hω,
we have (4.4)∥tn(s;t1)−s∥ω∗=O{(ω(πn+1))1−β/η+((n+1)−1∑k=1n+1ω(1k+1))1−β/η}.


### 5. Lemmas

We will use
following lemmas.Lemma 5.1.
Letϕt1(t) be defined in (2.16), then for s∈Hω,
we have (5.1)|ϕt1(t)−ϕt2(t)|≤4Kω(|t1−t2|),(5.2)|ϕt1(t)−ϕt2(t)|≤4Kω(|t|). It is easy to
verify.Lemma 5.2.
LetKn(t) be defined in (2.17), then (5.3)Kn(t)≤C(2n+1t)cosn(t2)sin(n+1)(t2), where “C" is an absolute constant, not
necessarily the same at each occurrence.Proof.
(5.4)Kn(t)=1sin(t/2)I.P.{∑k=0n(nk)ei(k+1/2)t}=1sin(t/2)I.P.{eit/2(1+eit)n}=1sin(t/2)I.P.{2ncosn(t2)ei(n+1)t/2}≤C(2n+1t)cosn(t2)sin(n+1)(tn).Lemma 5.3.
(5.5)∑k=0n(1t)cosk(t2)sin(k+1)(t2)≤(Ct2)(1−cos(n+1)(t2)cosn+1(t2)).Proof.
(5.6)∑k=0n(1t)cosk(t2)sin(k+1)(t2)=∑k=0n(1t)I.P.{ei(k+1)t/2cosk(t2)}=(1t)I.P.{eit/2(1−ei(n+1)t/2cosn+1(t/2)1−eit/2cos(t/2))}≤(Ct2)I.P.{i−icos(n+1)(t2)cosn+1(t2)+sin(n+1)(t2)cosn+1(t2)}=(Ct2)(1−cos(n+1)(t2)cosn+1(t2)).Lemma 5.4 (see [9]).
For0≤t≤1/n+1,
then (5.7)1−cos(n+1)(t2)cosn+1(t2)=O{(n+1)2t2}.Lemma 5.5 (see [6]).
Ifω(t) satisfies conditions (4.1) and (4.2), then (5.8)∫0ut−1ω(t)dt=O(uH(u)),u→0+.


### 6. Proof of Theorem4.1

Proof of Theorem4.1.
Following Zygmund [11],
we have (6.1)sn(t1)−s=12π∫0πϕt1(t)sin(t/2)sin(n+12)tdt.From
(2.6) and (2.16), we have (6.2)En1(t1)−s=2−n2π∫0πϕt1(t)Kn(t)dt.Using
Lemma 5.2, we have (6.3)En1(t1)−s≤C2−(n+1)π∫0πϕt1(t)t2n+1cosn(t2)sin(n+1)(t2)dt.Now
from (2.8), the (C,1)-transform of (E,1)-transform is given by (6.4)|tn(s;t1)−s|≤Cn+1∫0π|ϕt1(t)|t|∑k=0ncosk(t2)sin(k+1)(t2)|dt. Setting (6.5)En(t1)=|tn(s;t1)−s(t1)|≤Cn+1∫0π|ϕt1(t)|t|∑k=0ncosk(t2)sin(k+1)(t2)|dt,En(t1,t2)=|En(t1)−En(t2)|≤Cn+1∫0π|ϕt1(t)−ϕt2(t)|t|∑k=0ncosk(t2)sin(k+1)(t2)|dt=O(1n+1)(∫0π/n+1+∫π/n+1π)=I1+I2,say, now using
(4.1), (4.2), (5.2), and Lemma 5.5, we get (6.6)I1=O(1)1n+1∫0π/n+1t−1ω(t)dt=O{(n+1)−1H(πn+1)}. Again using (5.2), (4.1), and Lemma 5.3, we have (6.7)I2=O(1)1n+1∫π/n+1πt−2ω(t)|1−cos(n+1)(t2)cosn+1(t2)|dt=O(1)1n+1∫π/n+1πt−2ω(t)dt=O{(n+1)−1H(πn+1)}. Now
from (5.1), Lemmas 5.3 and 5.4, we have (6.8)I1=O(1)1n+1∫0π/n+1ω(|t1−t2|)t2|1−cos(n+1)(t2)cosn+1(t2)|dt=O(1)ω(|t1−t2|)n+1∫0π/n+1t−2(n+1)2t2dt=O{ω(|t1−t2|)},(6.9)I2=O(1)ω(|t1−t2|)n+1∫π/(n+1)πt−2dt=O{ω(|t1−t2|)}. Now noting that (6.10)Ir=Ir1−β/ηIrβ/η,r=1,2, we have, from (6.6) and (6.8), (6.11)I1=O{(ω(|t1−t2|))β/η((n+1)−1H(πn+1))1−β/η}, and from (6.7) and (6.9), we have (6.12)I2=O{(ω(|t1−t2|))β/η((n+1)−1H(πn+1))1−β/η}.  Thus, from (2.13), (6.11) and (6.12), we have (6.13)supt1,t2Δω∗|En(t1,t2)|=supt1,t2|En(t1)−En(t2)|ω∗(|t1−t2|)=O{(ω(|t1−t2|))β/η(ω∗(|t1−t2|))−1((n+1)−1H(πn+1))1−β/η}. It
is to be noted from (6.6) and (6.7), (6.14)∥En(t1)∥c=max0≤t1≤2π|tn(s;t1)−s|=O{(n+1)−1H(πn+1)}.  Combining (6.13) and (6.14), we get (6.15)∥tn(s;t1)−s∥ω∗=O{((n+1)−1H(πn+1))1−β/η}.This
completes the proof of Theorem 4.1.Proof of Theorem4.2.
Follows analogously as the proof of Theorem4.1 with slight changes, so
we omit details.


### 7. Applications

The following results can easily be derived
from the Theorem4.1. If we put ω∗(|t1−t2|)≤K|t1−t2|β, ω(|t1−t2|)≤A|t1−t2|α and replace η by α and set (7.1)H(u)={uα−1,0<α<1log(1u),α=1, then
we get Corollary 7.1.Corollary 7.1.
Ifs∈Hα, 0≤β<α≤1,
then (7.2)∥tn(s;t1)−s∥β={O(n+1)β−α,0<α<1O(log(n+1)(n+1))1−β,α=1. If we
put β=0,
then from above corollary, we have Corollary 7.2.Corollary 7.2.
Ifs∈Lipα, 0<α≤1,
then (7.3)∥tn(s;t1)−s∥={O(n−α),0<α<1O(lognn),α=1. Hence
Theorem 3 is particular case of Theorem 4.1.


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

(2.10)tn(s;t)=1n+1∑k=0nEk1=1n+1∑k=0n{12k∑υ=0k(kυ)sk}=∑k=0∞Cn,ksk, where(2.11)Cn,k={1n+12−k∑υ=0k(kυ),k≤n0,k>n.Now,(i)
∑k=0∞|Cn,k|=∑k=0n|(1/(n+1))2−k∑υ=0k(kυ)|=1,(ii)
Cn,k=(1/(n+1))(1)→0, as n→∞,
for fixed k,(iii)
∑k=0∞Cn,k=1,
thus, (C,1)(E,1)-method is regular.Singh [4] defined the space Hω by(2.12)Hω={s(t)∈C2π:|s(t1)−s(t2)|≤Kω(|t1−t2|)}, and
the norm ∥⋅∥ω∗ by (2.13)∥s∥ω∗=∥s∥c+Supt1,t2{Δω∗s(t1,t2)}, where (2.14)∥s∥c=Sup0≤t≤2π|s(t)|,Δω∗s(t1,t2)=|s(t1)−s(t2)|ω∗(|t1−t2|),t1≠t2,and
choosing Δ0s(t1,t2)=0, ω(t) 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 (2.15)Hα={s(t)∈C2π:|s(t1)−s(t2)|≤K|t1−t2|α,0<α≤1} is Banach
space [2] and the metric induced by the norm ∥⋅∥α on Hα is said to be Hölder metric.We write(2.16)ϕt1(t)=s(t1+t)+s(t1−t)−2s(t1),(2.17)Kn(t)=sin(n+1)t2∑k=0n(nk)sin(k+12)t.


## 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 functions:R→R is 2π-periodic and
belonging to class Lipα, 0<α≤1,
then the degree of approximation by
(C,1)(E,1) means of its Fourier series is given by(3.1)∥tn(s;t1)−s(t1)∥∞={O(n−α),0<α<1O(lognn),α=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 signals(t) and the processed output
signal tn(s;t)=(1/(n+1))∑k=1nEk1(t),
by establishing the following theorems.Theorem 4.1.
Letω(t) defined in (2.12) be such that (4.1)∫tπω(u)u2du=O{H(t)},H(t)≥0,(4.2)∫0tH(u)du=O{tH(t)},ast→0+, then, for 0≤β<η≤1 and s∈Hω,
we have (4.3)∥tn(s;t1)−s∥ω∗=O{((n+1)−1H(πn+1))1−β/η}.Theorem 4.2.
Letω(t) defined in (2.12) and for 0≤β<η≤1 and s∈Hω,
we have (4.4)∥tn(s;t1)−s∥ω∗=O{(ω(πn+1))1−β/η+((n+1)−1∑k=1n+1ω(1k+1))1−β/η}.


## 5. Lemmas

We will use
following lemmas.Lemma 5.1.
Letϕt1(t) be defined in (2.16), then for s∈Hω,
we have (5.1)|ϕt1(t)−ϕt2(t)|≤4Kω(|t1−t2|),(5.2)|ϕt1(t)−ϕt2(t)|≤4Kω(|t|). It is easy to
verify.Lemma 5.2.
LetKn(t) be defined in (2.17), then (5.3)Kn(t)≤C(2n+1t)cosn(t2)sin(n+1)(t2), where “C" is an absolute constant, not
necessarily the same at each occurrence.Proof.
(5.4)Kn(t)=1sin(t/2)I.P.{∑k=0n(nk)ei(k+1/2)t}=1sin(t/2)I.P.{eit/2(1+eit)n}=1sin(t/2)I.P.{2ncosn(t2)ei(n+1)t/2}≤C(2n+1t)cosn(t2)sin(n+1)(tn).Lemma 5.3.
(5.5)∑k=0n(1t)cosk(t2)sin(k+1)(t2)≤(Ct2)(1−cos(n+1)(t2)cosn+1(t2)).Proof.
(5.6)∑k=0n(1t)cosk(t2)sin(k+1)(t2)=∑k=0n(1t)I.P.{ei(k+1)t/2cosk(t2)}=(1t)I.P.{eit/2(1−ei(n+1)t/2cosn+1(t/2)1−eit/2cos(t/2))}≤(Ct2)I.P.{i−icos(n+1)(t2)cosn+1(t2)+sin(n+1)(t2)cosn+1(t2)}=(Ct2)(1−cos(n+1)(t2)cosn+1(t2)).Lemma 5.4 (see [9]).
For0≤t≤1/n+1,
then (5.7)1−cos(n+1)(t2)cosn+1(t2)=O{(n+1)2t2}.Lemma 5.5 (see [6]).
Ifω(t) satisfies conditions (4.1) and (4.2), then (5.8)∫0ut−1ω(t)dt=O(uH(u)),u→0+.


## 6. Proof of Theorem4.1

Proof of Theorem4.1.
Following Zygmund [11],
we have (6.1)sn(t1)−s=12π∫0πϕt1(t)sin(t/2)sin(n+12)tdt.From
(2.6) and (2.16), we have (6.2)En1(t1)−s=2−n2π∫0πϕt1(t)Kn(t)dt.Using
Lemma 5.2, we have (6.3)En1(t1)−s≤C2−(n+1)π∫0πϕt1(t)t2n+1cosn(t2)sin(n+1)(t2)dt.Now
from (2.8), the (C,1)-transform of (E,1)-transform is given by (6.4)|tn(s;t1)−s|≤Cn+1∫0π|ϕt1(t)|t|∑k=0ncosk(t2)sin(k+1)(t2)|dt. Setting (6.5)En(t1)=|tn(s;t1)−s(t1)|≤Cn+1∫0π|ϕt1(t)|t|∑k=0ncosk(t2)sin(k+1)(t2)|dt,En(t1,t2)=|En(t1)−En(t2)|≤Cn+1∫0π|ϕt1(t)−ϕt2(t)|t|∑k=0ncosk(t2)sin(k+1)(t2)|dt=O(1n+1)(∫0π/n+1+∫π/n+1π)=I1+I2,say, now using
(4.1), (4.2), (5.2), and Lemma 5.5, we get (6.6)I1=O(1)1n+1∫0π/n+1t−1ω(t)dt=O{(n+1)−1H(πn+1)}. Again using (5.2), (4.1), and Lemma 5.3, we have (6.7)I2=O(1)1n+1∫π/n+1πt−2ω(t)|1−cos(n+1)(t2)cosn+1(t2)|dt=O(1)1n+1∫π/n+1πt−2ω(t)dt=O{(n+1)−1H(πn+1)}. Now
from (5.1), Lemmas 5.3 and 5.4, we have (6.8)I1=O(1)1n+1∫0π/n+1ω(|t1−t2|)t2|1−cos(n+1)(t2)cosn+1(t2)|dt=O(1)ω(|t1−t2|)n+1∫0π/n+1t−2(n+1)2t2dt=O{ω(|t1−t2|)},(6.9)I2=O(1)ω(|t1−t2|)n+1∫π/(n+1)πt−2dt=O{ω(|t1−t2|)}. Now noting that (6.10)Ir=Ir1−β/ηIrβ/η,r=1,2, we have, from (6.6) and (6.8), (6.11)I1=O{(ω(|t1−t2|))β/η((n+1)−1H(πn+1))1−β/η}, and from (6.7) and (6.9), we have (6.12)I2=O{(ω(|t1−t2|))β/η((n+1)−1H(πn+1))1−β/η}.  Thus, from (2.13), (6.11) and (6.12), we have (6.13)supt1,t2Δω∗|En(t1,t2)|=supt1,t2|En(t1)−En(t2)|ω∗(|t1−t2|)=O{(ω(|t1−t2|))β/η(ω∗(|t1−t2|))−1((n+1)−1H(πn+1))1−β/η}. It
is to be noted from (6.6) and (6.7), (6.14)∥En(t1)∥c=max0≤t1≤2π|tn(s;t1)−s|=O{(n+1)−1H(πn+1)}.  Combining (6.13) and (6.14), we get (6.15)∥tn(s;t1)−s∥ω∗=O{((n+1)−1H(πn+1))1−β/η}.This
completes the proof of Theorem 4.1.Proof of Theorem4.2.
Follows analogously as the proof of Theorem4.1 with slight changes, so
we omit details.


## 7. Applications

The following results can easily be derived
from the Theorem4.1. If we put ω∗(|t1−t2|)≤K|t1−t2|β, ω(|t1−t2|)≤A|t1−t2|α and replace η by α and set (7.1)H(u)={uα−1,0<α<1log(1u),α=1, then
we get Corollary 7.1.Corollary 7.1.
Ifs∈Hα, 0≤β<α≤1,
then (7.2)∥tn(s;t1)−s∥β={O(n+1)β−α,0<α<1O(log(n+1)(n+1))1−β,α=1. If we
put β=0,
then from above corollary, we have Corollary 7.2.Corollary 7.2.
Ifs∈Lipα, 0<α≤1,
then (7.3)∥tn(s;t1)−s∥={O(n−α),0<α<1O(lognn),α=1. Hence
Theorem 3 is particular case of Theorem 4.1.



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*Source: 495075-2008-04-13.xml*