Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 298923, 13 pages
doi:10.1155/2012/298923
Research Article
Product Summability Transform of Conjugate
Series of Fourier Series
Vishnu Narayan Mishra,1 Kejal Khatri,1
and Lakshmi Narayan Mishra2
1
Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology,
Ichchhanath Mahadev Road, Surat 395 007, India
2
Dr. Ram Manohar Lohia Avadh University, Hawai Patti Allahabad Road, Faizabad 224 001, India
Correspondence should be addressed to Kejal Khatri, kejal0909@gmail.com
Received 22 February 2012; Revised 15 March 2012; Accepted 15 March 2012
Academic Editor: Ram U. Verma
Copyright q 2012 Vishnu Narayan Mishra et al. 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.
A known theorem, Nigam 2010 dealing with the degree of approximation of conjugate of a
signal belonging to Lipξt-class by E, 1C, 1 product summability means of conjugate series of
Fourier series has been generalized for the weighted WLr , ξt, r ≥ 1, t > 0-class, where ξt
1 1
C which is in more general form of Theorem 2
is nonnegative and increasing function of t, by E
of Nigam and Sharma 2011.
n
n
1. Introduction
Khan 1, 2 has studied the degree of approximation of a function belonging to Lipα, r and
WLr , ξt classes by Nörlund and generalized Nörlund 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 WLr , ξt class by linear operators.
Thereafter, Nigam 8 and Nigam and Sharma 9 discussed the degree of approximation of
conjugate of a function belonging to Lipξt, r class and WLr , ξt by E, 1C, 1 product
summability means, respectively. Recently, Rhoades et al. 10 have determined very interesting result on the degree of approximation of a function belonging to Lipα class by Hausdorff
means. Summability techniques were also applied on some engineering problems like Chen
and Jeng 11 who implemented the Cesàro sum of order C, 1 and C, 2, 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 Cesàro sum technique for the derivative of the double layer potential. Similarly, Chen
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International Journal of Mathematics and Mathematical Sciences
and Hong 13 used Cesàro sum regularization technique for hypersingularity of dual
integral equation.
The generalized weighted WLr , ξt, r ≥ 1-class is generalization of Lipα, Lipα, r
and Lipξt, r classes. Therefore, in the present paper, a theorem on degree of approximation
of conjugate of signals belonging to the generalized weighted WLr , ξt, r ≥ 1 class by
E, 1C, 1 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 fx be a 2π-periodic function and integrable in the sense of Lebesgue. The Fourier
series of fx at any point x is given by
∞
∞
a0 1.1
an cos nx bn sin nx ≡
An x,
fx ∼
2 n1
n0
with nth partial sumsn f; x.
The conjugate series of Fourier series 1.1 is given by
∞
∞
Bn x.
bn cos nx − an sin nx ≡
n1
1.2
n1
Let ∞
n0 un be a given infinite series with sequence of its nth partial sums {sn }. The E, 1
transform is defined as the nth partial sum of E, 1 summability, and we denote it by En1 .
If
n 1 n
En1 sk −→ s, as n −→ ∞,
1.3
2n k0 k
then the infinite series
If
τn ∞
n0
un is summable E, 1 to a definite number s, Hardy 14.
n
1 s0 s1 s2 · · · sn
sk −→ s,
n1
n 1 k0
as n −→ ∞,
1.4
then the infinite series ∞
n0 un is summable to the definite number s by C, 1 method. The
E, 1 transform of the C, 1 transform defines E, 1C, 1 product transform and denotes it
by En1 Cn1 . Thus, if
n 1 n 1
1 1
En Cn Ck −→ s, as n −→ ∞,
1.5
2n k0 k
then the infinite series ∞
n0 un is said to be summable by E, 1C, 1 method or summable
E, 1C, 1 to a definite number s. The E, 1 is regular method of summability
n
1 sk −→ s, as n −→ ∞, Cn1 method is regular,
sn −→ s ⇒ Cn1 sn τn n 1 k0
1.6
⇒ En1 Cn1 sn En1 Cn1 −→ s, as n −→ ∞, En1 method is regular,
⇒ En1 Cn1 method is regular.
International Journal of Mathematics and Mathematical Sciences
3
A function fx ∈ Lipα, if
fx t − fx O|tα |
for 0 < α ≤ 1, t > 0,
1.7
and fx ∈ Lipα, r, for 0 ≤ x ≤ 2π, if
ωr t; f 2π
fx t − fxr dx
1/r
O |t|α ,
0 < α ≤ 1, r ≥ 1, t > 0.
1.8
0
Given a positive increasing function ξt, fx ∈ Lipξt, r, if
ωr t; f 2π
fx t − fxr dx
1/r
Oξt,
r ≥ 1, t > 0.
1.9
0
Given positive increasing function ξt, an integer r ≥ 1, f ∈ WLr , ξt, 2, if
ωr t; f 2π
0
r
fx t − fx sinβ x dx
1/r
Oξt,
β ≥ 0 , t > 0.
1.10
For our convenience to evaluate I2 without error, we redefine the weighted class as follows:
ωr t; f 2π
0
fx t − fxr sinβr x dx
2
1/r
Oξt,
β ≥ 0, t > 0 16.
1.11
If β 0, then the weighted class WLr , ξt coincides with the class Lipξt, r, we observe
that
β0
ξttα
r →∞
WLr , ξt −→ Lipξt, r −→ Lipα, r −→ Lipα for 0 < α ≤ 1, r ≥ 1, t > 0.
1.12
Lr -norm of a function is defined by
f r
2π
fxr dx
1/r
,
r ≥ 1.
1.13
0
A signal f is approximated by trigonometric polynomials τn of order n, and the degree of
approximation En f is given by Rhoades 3
En f min fx − τn f; x r ,
n
1.14
in terms of n, where τn f; x is a trigonometric polynomial of degree n. This method of
approximation is called trigonometric Fourier approximation TFA 4.
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International Journal of Mathematics and Mathematical Sciences
We use the following notations throughout this paper:
ψt fx t − fx − t,
n
k
cosv
1/2t
1
1
n
.
G
n t n1
2 π k0 k 1 k v0 sint/2
1.15
2. Previous Result
conjugate
Nigam 8 has proved a theorem on the degree of approximation of a function fx,
to a periodic function fx with period 2π and belonging to the class Lipξt, r r ≥ 1, by
E, 1C, 1 product summability means of conjugate series of Fourier series. He has proved
the following theorem.
conjugate to a 2π-periodic function fx, belongs to Lipξt, r
Theorem 2.1 see 8. If fx,
class, then its degree of approximation by E, 1C, 1 product summability means of conjugate series
of Fourier series is given by
1
En1 Cn1 − f O n 11/r ξ
,
n1
r
2.1
provided ξt satisfies the following conditions:
1/n1 r
tψt
ξt
0
π
1/n1
1/r
dt
O
1
,
n1
r 1/r
t−δ ψt
dt
O n 1δ ,
ξt
2.2
2.3
where δ is an arbitrary number such that s1 − δ − 1 > 0, r −1 s−1 1, 1 ≤ r ≤ ∞, conditions 2.2
1 1
and 2.3 hold uniformly in x and E
C is E, 1C, 1 means of the series 1.2.
n
n
1 1
Remark 2.2. The proof proceeds by estimating |E
n Cn − f|, which is represented in terms of
an integral. The domain of integration is divided into two parts—from 0, 1/n 1 and
1/n 1, π. Referring to second integral as I2 , and using Hölder inequality, the author 8
obtains
|I2 | ≤
π
1/n1
O n 1δ
t
r
ψt
ξt
⎞ ⎫1/s
s ⎪
⎬
ξtG
n t
⎜
⎟
dt
⎝
⎠
⎪
⎪
t−δ
⎭
⎩ 1/n1
⎧
1/r ⎪ π
⎨
−δ dt
⎛
⎞ ⎫1/s
s ⎪
⎬
ξtG
n t
⎜
⎟
dt
.
⎝
⎠
⎪
⎪
t−δ
⎭
⎩ 1/n1
⎧
⎨ π
⎪
⎛
2.4
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5
The author then makes the substitution y 1/t to obtain
n1 ξ1/y s dy 1/s
δ
.
O n 1
y2
yδ−1
1/π
2.5
In the next step ξ1/y is removed from the integrand by replacing it with Oξ1/n 1,
while ξt is an increasing function, ξ1/y is now a decreasing function. Therefore, this step
is invalid.
1
− f|,
in Theorem 2 of Nigam and Sharma
Remark 2.3. The proof follows by obtaining |EC
n
9, which is expressed in terms of an integral. The domain of integration is divided into two
parts—from 0, 1/n 1 and 1/n 1, π. Referring to second integral as I2.2 , and using
Hölder inequality, the authors 9 obtain the following:
|I2.2 | ≤
π
1/n1
⎧
⎛
⎞ ⎫1/s
r 1/r ⎪ π
s ⎪
⎬
⎨
ξtG
n t
t−δ ψtsinβ t
⎟
⎜
dt
⎝ −δ β ⎠ dt
⎪
⎪
ξt
t sin t
⎭
⎩ 1/n1
π
δ
O n 1
1/n1
s 1/s
dt
.
1−δβ
ξt
t
2.6
2.7
The authors then make the substitution y 1/t to get
n1 ξ1/y s dy 1/s
δ
.
O n 1
y2
yδ−1−β
1/π
2.8
In the next step, ξ1/y is removed from the integrand by replacing it with Oξ1/n 1,
while ξt is an increasing function, ξ1/y is now a decreasing function. Therefore, in view
of second mean value theorem of integral, this step is invalid.
Remark 2.4. The condition 1/sinβ t O1/tβ , 1/n 1 ≤ t ≤ π used by Nigam and Sharma
9 is not valid, since sin t → 0 as t → π.
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 17–20 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.
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International Journal of Mathematics and Mathematical Sciences
conjugate to a 2π-periodic function f, belongs to the generalized weighted
Theorem 3.1. If fx,
WLr , ξtr ≥ 1-class, then its degree of approximation by E, 1C, 1 product summability means
of conjugate series of Fourier series is given by
1
En1 Cn1 − f O n 1β1/r ξ
,
n1
r
3.1
provided ξt satisfies the following conditions:
π/n1 r
1/r
tψt
1
t
βr
dt
,
sin
O
ξt
2
n1
0
π
π/n1
ξt
t
r 1/r
t−δ ψt
dt
On 1δ ,
ξt
is nonincreasing sequence in “t
,
3.2
3.3
3.4
where δ is an arbitrary number such that s1 − δ − 1 > 0, r −1 s−1 1, 1 ≤ r ≤ ∞, conditions 3.2
1 1
and 3.3 hold uniformly in x and E
n Cn is E, 1C, 1 means of the series 1.2 and the conjugate
function fx
is defined for almost every x by
1
−
fx
2π
π
0
1 π
t
t
dt limh → 0 −
dt .
ψt cot
ψt cot
2
2π h
2
3.5
Note 1. ξπ/n 1 ≤ πξ1/n 1, for π/n 1 ≥ 1/n 1.
Note 2. Also for β 0, 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 E, 1C, 1 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 |G
n t| O1/t for 0 < t ≤ π/n 1.
International Journal of Mathematics and Mathematical Sciences
7
Proof. For 0 < t ≤ π/n 1, sint/2 ≥ t/π and | cos nt| ≤ 1.
Gn t ≤
n
k
1 cosv 1/2t 1
n
1 k v0 sint/2
2π2n k0 k
k
n 1 1
|cosv 1/2t|
n
1 k v0 |sint/2|
2π2n k0 k
k
n
1 1 n
1
2t2n k0 k k 1 v0
n ' (
1 n
n
2t2 k0 k
' (
n 1
1
n
n
,
since
O
2n .
2
k
t
2t2n
k0
4.1
This completes the proof of Lemma 4.1.
Lemma 4.2. Consider |G
n t| O1/t, for 0 < t ≤ π and any n.
Proof. For 0 < π/n 1 ≤ t ≤ π, sint/2 ≥ t/π.
n
k
1 cosv 1/2t 1
n
Gn t 1 k v0 sint/2
2π2n k0 k
n
k
1 1
n
≤
Re
eiv1/2t n
1k
2t2 k0 k
v0
n
k
1 1
n
it/2 ivt
≤
Re
e
e
1k
2t2n k0 k
v0
n
k
1 1
n
ivt
≤
e
Re
1k
2t2n k0 k
v0
τ−1
k
1 1
n
ivt
≤
Re
e
n
k
1
k
2t2 k0
v0
n
k
1 1
n
ivt
Re
e
.
n
k
1
k
2t2 kτ
v0
4.2
Now considering first term of 4.2
τ−1
k
τ−1
k
1 1 1
1
n
n
ivt
Re
e
1 eivt ≤
n
n
k
1k
1
k
2t2 k0 k
2t2
v0
v0
k0
' (
τ−1
1 n ≤
.
2t2n k0 k 4.3
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International Journal of Mathematics and Mathematical Sciences
Now considering second term of 4.2 and using Abel’s lemma
n
k
m
n 1
1 1
1
n
n
eivt ≤
Re
max eivt n
n
k
1k
1
k
0≤m≤k
2t2 kτ k
2t2
v0
v0
kτ
(
n ' 1 1
n
≤
1 k
1k
2t2n kτ k
4.4
n ' (
1 n
.
n
2t2 kτ k
On combining 4.2, 4.3, and 4.4
Gn t ≤
τ−1 ' (
n 1 1 n
n
,
2t2n k0 k
2t2n kτ k
' (
1
.
Gn t O
t
4.5
This completes the proof of Lemma 4.2.
5. Proof of Theorem
Let s
n x denotes the nth partial sum of series 1.2. Then following Nigam 8, we have
cosn 1/2t
1 π
5.1
dt.
ψt
−
fx
s
x
n
2π 0
sint/2
Therefore, using 1.2, the C, 1 transform Cn1 of sn is given by
π
n
cosn 1/2t
1
)1 − fx
C
dt.
ψt
n
2πn 1 0
sint/2
k0
1
1
Now denoting E,
1C, 1 transform of sn as E
n , Cn , we write
k
n π
ψt
1
1 n
1
1
cosv 1/2t dt
En , Cn − fx 2π2n k0 k
0 sint/2 1 k
v0
π
ψtG
n tdt
0
π/n1 π
0
π/n1
I1 I2 say.
We consider, |I1 | ≤
* π/n1
0
|ψt||G
n t|dt.
ψtG
n tdt
5.2
5.3
International Journal of Mathematics and Mathematical Sciences
Using Hölder’s inequality
|I1 | ≤
π/n1 r
tψt
ξt
0
9
⎛
⎡
⎞s ⎤1/s
1/r π/n1 ξtG
t
⎟ ⎥
n
t
⎜
⎢
βr
dt
sin
⎝
⎠ dt⎦
⎣
β
2
tsin t/2
0
⎛
⎡
⎞s ⎤1/s
π/n1 ξtG
t
⎟ ⎥
n
1
⎜
⎢
O
⎝
⎠ dt⎦
⎣
n1
tsinβ t/2
0
by 3.2
π/n1 s 1/s
ξt
dt
by Lemma 4.1
t2 sinβ t/2
0
s 1/s
π/n1 ξtt/2β
1
O
dt
n1
t2 sinβ t/2 · t/2β
0
1/s
1
2π/2n 1 βs π/n1 ξt s
O
dt
, as h −→ 0
n1
sinπ/2n 1
t2β
h
π/n1 1/s
1
ξt s
O
dt
.
n1
t2β
h
1
O
n1
5.4
Since ξt is a positive increasing function and by using second mean value theorem for
integrals, we have
s 1/s
π/n1 π
1
1
π
I1 O
dt
, for some 0 < ∈ <
ξ
2β
n1
n1
n1
t
∈
5.5
1/s
π/n1
1
1
O
t−βs−2s dt
.
πξ
n1
n1
∈
Note that ξπ/n 1 ≤ πξ1/n 1,
I1 O
⎤
⎡
−βs−2s1 π/n1 1/s
1
1
t
⎦
⎣
ξ
n1
n1
−βs − 2s 1
∈
'
(
1
1
O
ξ
n 1β2−1/s
n1
n1
' (
1
O ξ
n 1β1−1/s
n1
(
' 1
∵ r −1 s−1 1, 1 ≤ r ≤ ∞.
O ξ
n 1β1/r
n1
Now, we consider
|I2 | ≤
π
π/n1
ψtG
t
dt.
n
5.6
5.7
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International Journal of Mathematics and Mathematical Sciences
Using Hölder’s inequality
|I2 | ≤
π
π/n1
⎛
⎡
⎞ ⎤1/s
r 1/r π
s
ξtG
n t
t−δ sinβ t/2ψt
⎜
⎟ ⎥
⎢
dt
⎝ −δ β
⎠ dt⎦
⎣
ξt
t sin t/2
π/n1
⎡
π
⎢
O n 1δ ⎣
⎛
π
δ
O n 1
⎞ ⎤1/s
s
ξtG
n t
⎜
⎟ ⎥
⎝ −δ β
⎠ dt⎦
t
sin
t/2
π/n1
π/n1
π
ξt
−δ
t tsinβ t/2
s 1/s
dt
by 3.3
5.8
by Lemma 4.2
s 1/s
ξt
dt
t−δ1 sinβ t/2
π/n1
1/s
π
ξt s
δ
O n 1
dt
.
t−δβ1
π/n1
O n 1δ
Now putting t 1/y, we have
n1/π ξ1/y s dy 1/s
δ
I2 O n 1
.
y2
yδ−β−1
1/π
5.9
Since ξt is a positive increasing function, so ξ1/y/1/y is also a positive increasing
function and using second mean value theorem for integrals, we have
1/s
n1/π dy
n1
1
δ ξπ/n 1
≤η≤
O n 1
, for some
−βsδs2
π/n 1
π
π
y
η
⎧
n1/π ⎫1/s
⎨
⎬
−δs−2βs1
y
n1
1
1
O n 1δ1 ξ
≤1≤
, for some
⎭
⎩
n1
−δs − 2 βs 1
π
π
1
1
2n1/π 1/s
1
O n 1δ1 ξ
y−δs−1βs
1
n1
1
O n 1δ1 ξ
n 1−δ−1/sβ
n1
1
O ξ
n 1δ1−δ−1/sβ
n1
1
O ξ
∵ r −1 s−1 1, 1 ≤ r ≤ ∞.
n 1β1/r
n1
Combining I1 and I2 yields
1
En1 Cn1 − f O n 11/rβ ξ
.
n1
5.10
5.11
International Journal of Mathematics and Mathematical Sciences
11
Now, using the Lr -norm of a function, we get
2π r 1/r
En1 Cn1 − f En1 Cn1 − f dx
r
0
2π r 1/r
1
O
ξ
dx
n 1
n1
0
⎧
⎫
2π 1/r ⎬
⎨
1
O n 1β1/r ξ
dx
⎩
⎭
n1
0
O n 1
β1/r
β1/r
1
ξ
n1
5.12
.
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 ξt tα , 0 < α ≤ 1, then the weighted class WLr , ξt, r ≥ 1 reduces to the class
Lipα, r and the degree of approximation of a function fx
conjugate to a 2π-periodic function f
belonging to the class Lipα, r, is given by
1
En1 Cn1 − f O
.
6.1
n 1α−1/r
Proof. The result follows by setting β 0 in 3.1, we have
2π r 1/r
1
1/r
En1 Cn1 − f En1 Cn1 − f dx
O
1
ξ
n
n1
r
0
1
, r ≥ 1.
O
n 1α−1/r
6.2
Thus, we get
2π r 1/r
En1 Cn1 − f ≤
En1 Cn1 − f dx
O
0
This completes the proof of Corollary 6.1.
1
n 1α−1/r
,
r ≥ 1.
6.3
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International Journal of Mathematics and Mathematical Sciences
Corollary 6.2. If ξt tα for 0 < α < 1 and r → ∞ in Corollary 6.1, then f ∈ Lipα and
1
En1 Cn1 − f O
.
n 1α
6.4
Proof. For r ∞ in Corollary 6.1, we get
1 1
En1 Cn1 − f sup E
C
−
fx
x
n n
∞
0≤x≤2π
1
O
.
n 1α
6.5
Thus, we get
En1 Cn1 − f ≤
En1 Cn1 − f
∞
1 1
sup En Cn x − fx
0≤x≤2π
O
6.6
1
.
n 1α
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 ElZoheary, 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.
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