R E S E A R C H
Open Access
Approximation of functions belonging to
Lip(
ξ
(
t
),
r
)
class by
(
N
,
p
n
)(
E
,
q
)
summability
of conjugate series of Fourier series
Vishnu Narayan Mishra
1*, Kejal Khatri
1and Lakshmi Narayan Mishra
2*Correspondence:
vishnunarayanmishra@gmail.com 1Department of Mathematics, Sardar Vallabhbhai National Institute of Technology,
Ichchhanath Mahadev Road, Surat, Gujarat 395007, India
Full list of author information is available at the end of the article
Abstract
In this paper, a new theorem concerning the degree of approximation of the conjugate of a function belonging to Lip(
ξ
(t),r) class by (N,pn)(E,q) summability of its conjugate series of a Fourier series has been proved. Here the product of Euler (E,q) summability method and Nörlund (N,pn) method has been taken.MSC: Primary 42B05; 42B08; 40G05
Keywords: degree of approximation; Lip(
ξ
(t),r)-class of function; (N,pn)(E,q) product summability; conjugate Fourier series; Lebesgue integral1 Introduction
Khan [, ] has studied the degree of approximation of a function belonging toLip(α,r )-class by Nörlund means. Generalizing the results of Khan [, ], many interesting results have been proved by various investigators like Mittalet al. [–], Mittal, Rhoades and Mishra [], Mittal and Singh [], Rhoadeset al.[], Mishraet al.[, ] and Mishra and Mishra [] for functions of various classesLipα,Lip(α,r),Lip(ξ(t),r) andW(Lr,ξ(t)),
(r≥) by using various summability methods. But till now, nothing seems to have been done so far to obtain the degree of approximation of conjugate of a function using (N,pn)(E,q) product summability method of its conjugate series of Fourier series. In this
paper, we obtain a new theorem on the degree of approximation of a functionf˜, con-jugate to a periodic functionf ∈Lip(ξ(t),r)-class, by (N,pn)(E,q) product summability
means.
Let∞n=unbe a given infinite series with the sequence of itsnth partial sums{sn}. Let
{pn}be a non-negative generating sequence of constants, real or complex, and let us write
Pn= n
k=
pk= ∀n≥, p–= =P– and Pn→ ∞ asn→ ∞.
The conditions for regularity of Nörlund summability are easily seen to be
() limn→∞pPnn→and
() ∞k=|pk|=O(Pn), asn→ ∞.
The sequence-to-sequence transformation
tnN= Pn
n
k=
pn–ksk (.)
defines the sequence{tN
n} of Nörlund means of the sequence{sn}, generated by the
se-quence of coefficients{pn}. The series∞n=unis said to be summable (N,pn) to the sums
iflimn→∞tNn exists and is equal tos.
The (E,q) transform is defined as thenth partial sum of (E,q) summability, and we de-note it byEqn. If
Eqn= ( +q)n
n
k=
n k
qn–ksk→s asn→ ∞, (.)
then the infinite series ∞n=unis said to be summable (E,q) to the sumsHardy [].
The (N,pn) transform of the (E,q) transform defines (N,pn)(E,q) product transform and
denotes it bytNE
n . This is if
tnNE= Pn
n
k= pn–k
( +q)k k
ν=
k
ν
qk–νsν. (.)
IftNE
n →sasn→ ∞, then the infinite series
∞
n=unis said to be summable (N,pn)(E,q)
to the sums.
sn→s ⇒ (E,q)(sn) =Enq= ( +q)–n n
k=
k n
qn–ksk→s,
asn→ ∞, (E,q) method is regular,
⇒ (N,pn)(E,q)(sn)
=tNEn →s, asn→ ∞, (N,pn) method is regular,
⇒ (N,pn)(E,q) method is regular.
A functionf(x)∈Lipαif
f(x+t) –f(x) = Otα for <α≤,t>
andf(x)∈Lip(α,r), for ≤x≤πif
π
f(x+t) –f(x)rdx
/r
= O|t|α, <α≤,r≥,t> .
Given a positive increasing functionξ(t),f(x)∈Lip(ξ(t),r), [] if
ωr(t;f) =
π
f(x+t) –f(x)rdx
/r
= Oξ(t), r≥,t> , (.)
we observe that
Lipξ(t),r ξ(t)=t
α
Lr-norm of a functionf:R→Ris defined by
fr=
π
f(x)rdx
/r
, r≥. (.)
L∞-norm of a functionf :R→Ris defined byf∞=sup{|f(x)|:x∈R}.
A signal (function)f is approximated by trigonometric polynomialstnof ordernand
the degree of approximationEn(f) is given by Zygmund []
En(f) =min
n f(x) –tn(f;x)r (.)
in terms ofn, wheretn(f;x) is a trigonometric polynomial of degreen. This method of
approximation is called Trigonometric Fourier Approximation (TFA) [].
The degree of approximation of a functionf :R→Rby a trigonometric polynomialtn
of ordernunder sup norm ∞is defined by
tn–f∞=suptn(x) –f(x):x∈R
.
Letf(x) be a π-periodic function and Lebesgue integrable. The Fourier series off(x) is given by
f(x)∼a +
∞
n=
(ancosnx+bnsinnx)≡
∞
n=
An(x) (.)
withnth partial sumsn(f;x).
The conjugate series of Fourier series (.) is given by
∞
n=
(ansinnx–bncosnx)≡
∞
n=
Bn(x). (.)
Particular cases:
() (N,pn)(E,q)means reduces to(N,n+)(E,q)means ifpn=n+.
() (N,pn)(E,q)means reduces to(N,n+ )(E, )means ifpn=n+ andqn= ∀n.
() (N,pn)(E,q)means reduces to(N,pn)(E, )means ifqn= ∀n.
() (N,pn)(E,q)means reduces to(C,δ)(E,q)means ifpn=
n+δ–
δ–
,δ> . () (N,pn)(E,q)means reduces to(C,δ)(E, )means ifpn=
n+δ–
δ–
,δ> andqn= ∀n.
() (N,pn)(E,q)means reduces to(C, )(E, )means ifpn= andqn= ∀n.
We use the following notations throughout this paper:
ψ(t) =f(x+t) –f(x–t),
˜ Gn(t) =
πPn
n
k= pn–k
( +q)k k
v=
k
ν
qk–νcos(v+ /)t
sint/
.
2 Main result
the present paper is to establish a quite new theorem on the degree of approximation of a functionf˜(x), conjugate to a π-periodic functionf belonging toLip(ξ(t),r)-class, by (N,pn)(E,q) means of conjugate series of Fourier series. In fact, we prove the following
theorem.
Theorem . Iff˜(x)is conjugate to aπ-periodic function f belonging toLip(ξ(t),r)-class, then its degree of approximation by(N,pn)(E,q)product summability means of conjugate
series of Fourier series is given by
˜tNEn –f˜r= O
(n+ )/rξ
n+
(.)
providedξ(t)satisfies the following conditions:
π/n+
t|
ψ(t)|
ξ(t)
r
dt
/r
= O(n+ )– (.)
and
π
π/n+
t–δ|ψ(t)|
ξ(t)
r
dt
/r
= O(n+ )δ, (.)
whereδis an arbitrary number such that s( –δ) – > ,r–+s–= , ≤r≤ ∞,conditions (.)and(.)hold uniformly in x and˜tNEn is(N,pn)(E,q)means of the series(.),and the
conjugate functionf˜(x)is defined for almost every x by
˜
f(x) = – π
π
ψ(t)cot(t/)dt=lim
h→
– π
π
h
ψ(t)cot(t/)dt
. (.)
Note . ξ(nπ+)≤π ξ(n+ ), for (nπ+)≥(n+).
Note . The product transform plays an important role in signal theory as a double
digital filter [] and the theory of machines in mechanical engineering.
3 Lemmas
For the proof of our theorem, the following lemmas are required.
Lemma . | ˜Gn(t)|= O[/t]for <t≤π/(n+ ).
Proof For <t≤π/(n+ ),sin(t/)≥(t/π) and|cosnt| ≤,
G˜n(t)=
πPn
n
k=
pn–k
( +q)k k
v=
k
ν
qk–νcos(v+ /)t
sint/
≤ πPn
n
k=
pn–k
( +q)k k
v=
k
ν
qk–ν|cos(v+ /)t|
|sint/|
≤ tPn
n
k=
pn–k
( +q)k k
v=
k
ν
qk–ν
, since
k
ν=
k
ν
= tPn
n
k=
pn–k
( +q)k( +q) k
= tPn
n
k= pn–k
= O[/t], since
n
k=
pn–k=Pn.
This completes the proof of Lemma ..
Lemma . | ˜Gn(t)|= O[/t]for <π/(n+ )≤t≤πand any n.
Proof For <π/(n+ )≤t≤π,sin(t/)≥(t/π),
G˜n(t)=
πPn
n k=
pn–k
( +q)k k v= k ν
qk–νcos(v+ /)t
sint/
≤ tPn
n k=
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νei(v+/)t
≤ tPn
n k=
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νeivt
eit/
= tPn
n k=
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νeivt
= tPn
τ– k=
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νeivt
+ tPn
n
k=τ
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νeivt
. (.)
Now, considering the first term of equation (.),
tPn
τ– k=
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νeivt
≤ tPn
τ– k=
pn–k
( +q)k
k v= k ν
qk–ν
eivt
≤ tPn
τ– k=
pn–k
( +q)k
k v= k ν
qk–ν
= tPn
τ– k= pn–k
Now, considering the second term of equation (.) and using Abel’s lemma
tPn
n
k=τ
pn–k
( +q)k
k
v=
Real part of
k
ν
qk–νeivt
≤ tPn
n
k=τ
pn–k
( +q)kmax≤m≤k
m v= k ν
qk–νeivt
≤ tPn
n
k=τ
pn–k
( +q)kmax≤m≤k m v= k ν
qk–νeivt
= tPn
n
k=τ
pn–k
( +q)kmax≤m≤k m v= k ν
qk–ν
≤ tPn
n
k=τ
pn–k
( +q)k k v= k ν
qk–ν= tPn
n
k=τ
pn–k. (.)
On combining (.), (.) and (.), we have
G˜n(t)≤
tPn
τ–
k= pn–k+
tPn
n
k=τ
pn–k,
G˜n(t)= O[/t].
This completes the proof of Lemma ..
4 Proof of theorem
Let˜sn(x) denote the partial sum of series (.), we have
˜
sn(x) –f˜(x) =
π
π
ψ(t)cos(n+ /)t
sint/ dt.
Therefore, using (.), the (E,q) transformEnqof˜snis given by
˜
Eqn(x) –f˜(x) = π( +q)k
π
ψ(t)
sint/
n k= n k
qn–kcos(k+ /)t
dt.
Now, denoting(N,pn)(E,q) transform ofs˜nas˜tNEn , we write
˜
tnNE(x) –f˜(x) = πPn
n
k=
pn–k
( +q)k
π
ψ(t)
sint/
k ν= k ν
qk–νcos(ν+ /)t
dt = π
ψ(t)G˜n(t)dt
=
π/(n+)
+
π
π/(n+)
ψ(t)G˜n(t)dt
We consider
|I| ≤ π/(n+)
ψ(t)G˜n(t)dt.
Using Hölder’s inequality, equation (.) and Lemma (.), we get
|I| ≤ π/(n+)
t|ψ(t)|
ξ(t)
r
dt
/r
lim
h→
π/(n+)
h
ξ(t)| ˜Gn(t)|
t s dt /s = O n+
lim
h→
π/(n+)
h
ξ(t)| ˜Gn(t)|
t s dt /s = O n+
lim
h→
π/(n+)
h
ξ(t)
t
s
dt
/s
.
Sinceξ(t) is a positive increasing function, using the second mean value theorem for integrals,
I= O
n+
ξ
π
n+
lim
h→
π/(n+)
h t s dt /s = O n+
π ξ
n+
lim
h→
π/(n+)
h
t–sdt
/s
, in view of note (.)
= O
n+
ξ
n+
t–s+ –s+
π/(n+)
h
/s
, h→
= O
n+
ξ
n+
(n+ )–/s
= O ξ n+
(n+ )–/s
= O ξ n+
(n+ )/r
∵r–+s–= , ≤r≤ ∞. (.)
Now, we consider
|I| ≤ π
π/(n+)
ψ(t)G˜n(t)dt.
Using Hölder’s inequality, equation (.) and Lemma ., we have
|I| ≤
π
π/(n+)
t–δ|ψ(t)|
ξ(t)
r
dt
/r π
π/(n+)
ξ(t)| ˜Gn(t)|
t–δ
s
dt
/s
= O(n+ )δ
π
π/(n+)
ξ(t)| ˜Gn(t)|
t–δ
s
dt
/s
= O(n+ )δ
π
π/(n+)
ξ(t) t–δt
s
dt
/s
= O(n+ )δ π
π/(n+)
ξ(t) t–δ+
s
dt
/s
Now, puttingt= /y,
I= O
(n+ )δ (n+)/ π
/π
ξ(/y)
yδ–
s
dy y
/s
.
Sinceξ(t) is a positive increasing function, so ξ(//yy) is also a positive increasing function and using the second mean value theorem for integrals, we have
I= O
(n+ )δξ(π/n+ )
π/n+
(n+)/π
/π
dy yδs+
/s
= O
(n+ )δ+ξ
n+
y–δs–+ –δs– +
(n+)/π
/π
/s
= O
(n+ )δ+ξ
n+
y–δs–(/nπ+)/π/s
= O
(n+ )δ+ξ
n+
(n+ )–δ–/s
= O
ξ
n+
(n+ )δ+–δ–/s
= O
ξ
n+
(n+ )/r
∵r–+s–= , ≤r≤ ∞. (.)
CombiningIandIyields
˜tNEn –f˜= O
(n+ )/rξ
n+
. (.)
Now, using theLr-norm of a function, we get
˜tNEn –f˜r= π
˜tNEn –f˜rdx
/r
= O π
(n+ )/rξ
n+
r
dx
/r
= O
(n+ )/rξ
n+
π
dx
/r
= O
(n+ )/rξ
n+
.
This completes the proof of Theorem ..
5 Applications
The study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. The following corollaries can be derived from our main Theorem ..
Corollary . Ifξ(t) =tα, <α≤,then the classLip(ξ(t),r),r≥reduces to the class
π-periodic function f belonging to the classLip(α,r),by(N,pn)(E,q)-means is given by
˜tNEn –f˜= O
(n+ )α–/r
. (.)
Proof We have
˜tNEn –f˜r= π
˜tNEn (x) –f˜(x)rdx
/r
= O(n+ )/rξ/(n+ )
= O(n+ )–α+/r.
Thus, we get
˜tNEn –f˜≤ π
˜tnNE(x) –f˜(x)rdx
/r
= O(n+ )–α+/r, r≥.
This completes the proof of Corollary ..
Corollary . Ifξ(t) =tαfor <α< and r=∞in Corollary.,then f ∈Lipαand
˜tNEn –f˜= O
(n+ )α
. (.)
Proof Forr→ ∞, we get
˜tNEn –f˜∞= sup
≤x≤π
˜tNEn (x) –f˜(x)= O(n+ )–α.
Thus, we get
˜tNEn –f˜≤˜tNEn –f˜∞
= sup
≤x≤π
˜tnNE(x) –f˜(x)
= O(n+ )–α.
This completes the proof of Corollary ..
Corollary . Ifξ(t) =tα, <α≤,then the classLip(ξ(t),r),r≥,reduces to the class
Lip(α,r), /r<α≤and if q= , then(E,q)summability reduces to(E, ) summability and the degree of approximation of a functionf˜(x),conjugate to aπ-periodic function f belonging to the classLip(α,r),by(N,pn)(E, )-means is given by
˜tNEn –f˜r= O
(n+ )α–/r
. (.)
Corollary . Ifξ(t) =tαfor <α< and r=∞in Corollary.,then f ∈Lipαand
˜tNEn –f˜∞= O
(n+ )α
Remark An independent proof of above Corollary . can be obtained along the same line of our main theorem.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
VNM, KK and LNM contributed equally to this work. All the authors read and approved the final manuscript.
Author details
1Department of Mathematics, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat, Gujarat 395007, India.2Dr. Ram Manohar Lohia Avadh University, Hawai Patti Allahabad Road, Faizabad, Uttar Pradesh 224 001, India.
Acknowledgements
The authors take this opportunity to express their gratitude to prof. Huzoor Hasan Khan, Department of Mathematics, Aligarh Muslim University, Aligarh, for suggesting the problem as well as his valuable suggestions for the improvement of the paper. The authors would like to thank the anonymous referees for several useful interesting comments about the paper. The authors are also thankful to Prof. Ravi P. Agrawal, editor-in-chief of JIA, Texas A & M University, Kingsville. Special thanks are due to our great master and friend academician Prof. Vijay Gupta, Netaji Subhas Institute of Technology, New Delhi, for kind cooperation, smooth behavior during communication and for his efforts to send the reports of the manuscript timely.
Received: 6 March 2012 Accepted: 23 November 2012 Published: 13 December 2012 References
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Cite this article as:Mishra et al.:Approximation of functions belonging toLip(ξ(t),r)class by(N,pn)(E,q)summability
