Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 5 Issue 4 (2013), Pages 40-53
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS
(FUNCTIONS) BELONGING TO THE GENERALIZED
WEIGHTED LIPSCHITZ W (Lr , ξ(t)), (r ≥ 1) - CLASS BY (C, 1) (E,
Q) MEANS OF CONJUGATE TRIGONOMETRIC FOURIER
SERIES
(COMMUNICATED BY HÜSEYIN BOR)
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
Abstract. Very recently, Sonker and Singh [21] determined the degree of approximation of the conjugate of 2π-periodic signals (functions) belonging to
Lip(α, r)(0 < α ≤ 1, r ≥ 1)-class by using Cesàro-Euler (C,1) (E,q) means of
their conjugate trigonometric Fourier series. In the present paper, we generalize the result of Sonker and Singh [21] on the generalized weighted Lipschitz
W (Lr , ξ(t)), (r ≥ 1)- class of signals (functions) by product summability (C,1)
(E,q) transform of conjugate trigonometric Fourier series. Our result also
generalizes the result of Lal and Singh [6]. Few applications and example of
approximation of functions will also be highlighted.
1. Introduction
The degree of approximation of functions belonging to Lipα, Lip(α, r), Lip(ξ(t), r)
and W (Lr , ξ(t)), (r ≥ 1) - classes by general summability matrices has been proved
by various investigators like Govil [1], Khan [3-5], Mohapatra and Chandra [19],
Mittal et al. [8-9], Mittal and Mishra [7], Mishra et al. [10-17] and Mishra and
Mishra [18]. Recently, Sonker and Singh [21] discussed the degree of approximation
of the conjugate of signals (functions) belonging to Lip(α, r) class by (C, 1)(E, q)
means of conjugate trigonometric Fourier series. But nothing seems to have been
done so far to obtain the degree of approximation of conjugate of signals belonging
to the generalized weighted Lipschitz W (Lr , ξ(t)), (r ≥ 1) class by (C,1) (E, q)
product summability transforms. Weighted W (Lr , ξ(t)), (r ≥ 1) Lipschitz - class
is generalization of Lipα, Lip(α, r) and Lip(ξ(t), r) - classes. In the present paper, an attempt to make advance study in this direction, a theorem on the degree
of approximation of conjugate of signals (functions) belonging to the generalized
02000 Mathematics Subject Classification: 41A10, 42B05, 42B08.
Keywords and phrases. Degree of approximation, weighted Lipschitz W (Lr , ξ(t)), (r ≥ 1), (t > 0)
- class of functions, (E, q) transform, (C,1) transform, product summability (C,1) (E, q)
transform, conjugate Fourier series, Lebesgue integral.
c 2013 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted August 9, 2013. Published October 6, 2013.
40
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS)
41
weighted Lipschitz W (Lr , ξ(t)), r ≥ 1 class by product summability (C,1) (E, q)
transform of conjugate series of Fourier series has been established.
Let L2π be the space of all 2π - periodic and Lebesgue-integrable functions over
[−π, π].Then the Fourier series of f ∈ L2π at x is given by
f (x) ∼
∞
∞
k=1
k=0
X
a0 X
+
(ak cos kx + bk sin kx) ≡
Ak (x),
2
(1)
with nth partial sums sn (f ; x), called trigonometric polynomial of degree (or order)
n, of the first (n + 1) terms of the Fourier series of f , ak and bk are the Fourier
coefficients of f .
The conjugate series of Fourier series (1) is given by
∞
X
k=1
(bk cos kx − ak sin kx) ≡
∞
X
Bk (x).
(2)
k=1
P∞
Let n=0 un be a given infinite series with sequence of its nth partial sums {sn }.
Hausdorff matrix H ≡ (hn,k ) is an infinite lower triangular matrix defined by
n n−k
µk , 0 ≤ k ≤ n,
k ∆
hn,k =
0,
k > n,
where ∆ is the forward difference operator defined by ∆µn = µn − µn+1 and
∆k+1 µn = ∆k (∆µn ). If H is regular, then {µn }, known as moment sequence, has
the representation
Z 1
un dγ(u),
µn =
0
where γ(u) known as mass function, is continuous at u = 0 and belongs to BV[0,1]
such that γ(0) = 0, γ(1) = 1; and for 0 < u < 1, γ(u) = [γ(u + 0) + γ(u − 0)]/2.
The Hausdorff means of conjugate Fourier series of f are defined by
H̃n (f ; x) =
n
X
k=0
hn,k s̃k , n ≥ 0.
The conjugate Fourier series is said to be summable to s by Hausdorff means, if
H̃n (f ; x) → s, as n → ∞. For the mass function γ(u) is given by
0, 0 ≤ u < a,
γ(u) =
1, a ≤ u ≤ 1,
where a = 1/(1 + q), q > 0, we can verify that µk = 1/(1 + q)k , and
( n−k
n q
k (1+qn ) , 0 ≤ k ≤ n,
hn,k =
0,
k > n.
Thus Hausdorff matrix H ≡ (hn,k ) reduces to Euler matrix (E, q) of order q > 0.
One more example of Hausdorff matrix [γ(u) = u for 0 ≤ u ≤ 1] is the well known
Cesàro matrix of order 1 (C,1).
The (E, q) transform is defined as the nth partial sum of (E, q), q > 0 summability
and we denote it by Enq . If
n X
1
n
q
q n−k sk → s, as n → ∞,
(3)
En =
k
(q + 1)n
k=0
42
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
then the infinite series
P∞
n=0
un is summable (E, q) to a definite number s [2]. If
n
s0 + s1 + s2 + ... + sn
1 X
τn =
=
sk → s, as n → ∞,
n+1
n+1
(4)
k=0
P
then the infinite series ∞
n=0 un is summable to the definite number s by (C,1)
method. The (C,1) transform of the (E, q) transform defines (C,1) (E, q) product
transform and denote it by Enq Cn1 . Thus if
n
k X
1 X
k
q k−v sv → s, as n → ∞,
(5)
(1 + q)−k
Cn1 Enq =
v
n+1
v=0
k=0
P
then the infinite series ∞
n=0 un is said to be summable by (C,1) (E, q) method or
summable (C,1) (E, q) to a definite number ’s’. P
∞
If Cn1 Enq → s as n → ∞ then the infinite series n=0 un or the sequence {sn } is
said to be summable (C,1) (E, q) to the sum s if limn→∞ Cn1 Enq exists and is equal
to s.
n X
1
n
sn → s ⇒ (E, q)(sn ) = Enq =
q n−k sk → s, as n → ∞, (E, q) method is regular,
k
(1 + q)n
k=0
Cn1 Enq
⇒ (C, 1)((E, q)(sn )) =
→ s, as n → ∞, (C, 1) method is regular,
⇒ (C, 1)(E, q) method is regular.
A signal (function) is said to belong to the class Lipα, if
|f (x + t) − f (x)| = O(|tα |) f or 0 < α ≤ 1, t > 0.
(6)
and f (x) ∈ Lip(α, r), f or 0 ≤ x ≤ 2π , if
Z 2π
r1
= O |t|α , 0 < α ≤ 1, r ≥ 1, t > 0.
kf (x+t)−f (x)kr =
|f (x+t)−f (x)|r dx
0
(7)
For a given a positive increasing function ξ(t), f (x) ∈ Lip(ξ(t), r) if
Z 2π
r1
kf (x + t) − f (x)kr =
|f (x + t) − f (x)|r dx
= O ξ(t) , r ≥ 1, t > 0. (8)
0
Given positive increasing function ξ(t), an integer r ≥ 1, a signal (function) f is
said to belong to the generalized weighted Lipschitz W (Lr , (ξ(t)))-class ([5]), if
[f (x + t) − f (x)]sinβ x/2 = O ξ(t) , β ≥ 0, t > 0.
(9)
r
If β = 0, then the generalized weighted Lipschitz W (Lr , (ξ(t))) class coincides with
the class Lip(ξ(t), r), we observe that
β=0
ξ(t)=tα
r→∞
W (Lr , (ξ(t)) −−−→ Lip(ξ(t), r) −−−−−→ Lip(α, r) −−−→ Lip(α) f or 0 < α ≤ 1, r ≥ 1, t > 0.
(10)
The Lr - norm of signal f is defined by
Z 2π
1r
r
, (1 ≤ r < ∞), and kf k∞ = sup |f (x)|.
(11)
kf kr =
|f (x)| dx
0
x∈[0,2π]
A signal (function) f is approximated by trigonometric polynomials τn (f ) of order
n and the degree of approximation En (f ) is given by [22]
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS)
En (f ) = min kf (x) − τ (f ; x)kr ,
43
(12)
n
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) [8].
We note that Enq and Cn1 Enq are also trigonometric polynomials of degree (or order)
n.
The conjugate function fe(x) is defined for almost every x by
1
fe(x) = −
2π
Z
π
ψ(t)cos t/2 dt = lim
h→0
0
−
1
2π
Z
π
ψ(t)cos t/2 dt
h
(see [22, p. 131]).
(13)
We use the following notations throughout this paper
ψ(t) = ψx (t) = f (x + t) − f (x − t),


cos
v
+
1/2
t
n
k
X
X
1
1
k
k−v

e

q
Gn (t) =
.
2π(1 + n) 
(1 + q)k v=0 v
k=0
sin t/2
2. Known Theorem
In a recent paper, Sonker and Singh [21] have studied the degree of approximation
of functions belonging to Lip(α, r) - class using (C,1) (E, q) summability means of
its conjugate Fourier series. They proved the following theorem:
Theorem 2.1. Let f (x) be a 2π- periodic, Lebesgue integrable function and belong
to the Lip(α, r) - class with r ≥ 1 and αr ≥ 1. Then the degree of approximation of
fe(x), the conjugate of f (x) by (C,1) (E, q) means of its conjugate Fourier series is
given by
(14)
kC 1 E q − fekr = O n−α+1/r , n = 0, 1, 2, ...,
n
provided
Z
n
π/(n+1)
0
Z π
|ψ(t)|
tα
r r1
= O((n + 1)−1 ),
dt
r r1
−δ |ψ(t)|
= O((n + 1)δ ),
t
dt
tα
π/(n+1)
(15)
(16)
where δ is an arbitrary number such that (α+δ)s+1 < 0 and r−1 +s−1 = 1 f or r >
1.
3. Main Theorem
Approximation by trigonometric polynomials is at the heart of approximation
theory. The most important trigonometric polynomials used in the approximation
theory are obtained by linear summation methods of Fourier series of 2π -periodic
functions on the real line (i.e. Cesàro means, Nörlund means and Product CesàroNörlund means, Product Ceàsro-Euler means etc.). Much of the advance in the
theory of trigonometric approximation is due to the periodicity of the functions.
Recently, Mishra et al. [14] have studied the degree of approximation of functions
belonging to W (Lr , ξ(t)) - class through (E, q)(C, 1) means of conjugate Fourier
44
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
series. In this paper, we use the (C, 1)(E, q) means of conjugate Fourier series of
f ∈ W (Lr , ξ(t)) to determine the degree of approximation of the conjugate of f ,
which in turn generalizes the result of Sonker and Singh [21] and Lal and Singh [6].
More precisely, we prove
Theorem 3.1. If fe(x), conjugate to a 2π - periodic function f belonging to the
generalized weighted Lipschitz W (Lr , ξ(t)), (r ≥ 1) - class then its degree of approximation by (C,1) (E, q) product summability means of conjugate series of Fourier
series is given by
n√ 1 o 1 ^
1 E q − fek = O ( n) r +β ξ √
,
kC
n
r
n
n
(17)
provided ξ(t) satisfies the following conditions:
Z
π
√
n
0
Z
π
π
√
n
t|ψ(t)|
ξ(t)
r
t−δ |ψ(t)|
ξ(t)
sinβr t/2 dt
1/r
1
=O √ ,
n
r 1/r
δ
√
n ,
dt
=O
(18)
(19)
ξ(t)
and
t is non-increasing in ’t’,
where δ is an arbitrary number such that s(1−δ +β)−1 > 0, r1 + 1s = 1, 1 ≤ r ≤ ∞,
^
1 q
^
conditions (18) and (19) hold uniformly in x and C
n En is (C, 1)(E, q) means of the
series (2) and
1
f (x) = −
2π
Note 3.2 ξ
√π
n
≤ πξ
√1
n
Z
π
ψ(t)cos t/2dt.
(20)
0
, for √πn ≥ √1n .
Note 3.3.The product transform (C,1) (E, q) plays an important role in signal theory as a double digital filter [13] and theory of Machines in Mechanical Engineering
[13].
4. Lemmas
For the proof of our theorem, the following lemmas are required:
en (t)| = O
Lemma 4.1. |G
1
t
+ ((n + 1)t) f or 0 ≤ t ≤
√π
n
≤
π
√
.
v
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS)
Proof. For 0 ≤ t ≤
e n (t)|
|G
≤
=
≤
=
=
=
=
=
√π ,
n
sin 2t ≥
t
π
45
and |cos nt| ≤ 1.
n
cos v + 1/2 t k X
X
1
1
k
k−v
q
2π(n + 1) (1 + q)k v=0 v
k=0
sin t/2
n
cos v + 1 − 1/2 t k
X
X
1
1
k
k−v
q
2π(n + 1) (1 + q)k v=0 v
k=0
sin t/2
cos
v
+
1
t
cos
t/2
+
sin
v
+
1
t
sin
t/2 n
k
X
X
1
1
k
k−v q
(n + 1)
(1 + q)k v=0 v
k=0
sin t/2
cos v + 1 tcos t/2 + sin v + 1 t sin t/2 n
k X
X
1
1
k
k−v q
k
(n + 1)
(1 + q) v=0 v
k=0
sin t/2
n
k X
X
1
1
1
k
k−v
+ O sin(v + 1)t
O
q
(n + 1)
(1 + q)k v=0 v
t
k=0
"
#
"
#
n
n
k k X
X
X
X
1
1
1
1
k
k
k−v
k−v
O
+O
q
q
(v + 1)t
(n + 1)t
(1 + q)k v=0 v
(n + 1)
(1 + q)k v=0 v
k=0
k=0
1
1
(n + 1) + O
(n + 1)(n + 1)t
O
(n + 1)t
(n + 1)
1
O
+ O [(n + 1)t] ,
t
in view of sin(v + 1)t ≤ (v + 1)t for 0 ≤ t ≤
This completes the proof of Lemma (4.1)
en (t)| = O
Lemma 4.2. |G
1
t
+ O(1) f or
√π
n
√π
n
≤
≤
π
√
.
v
π
√
v
≤ t ≤ π.
46
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
Proof. For 0 ≤
√π
n
e n (t)|
|G
≤
=
≤
=
=
=
=
=
π
√
v
≤ t ≤ π, sin 2t ≥
t
π
and |cos nt| ≤ 1.
n
cos
v
+
1/2
t k
X k
X
1
1
k−v
q
2π(n + 1) (1 + q)k v=0 v
k=0
sin t/2
n
cos v + 1 − 1/2 t k X
X
1
1
k
k−v
q
2π(n + 1) (1 + q)k v=0 v
k=0
sin t/2
cos
v
+
1
tcos
t/2
+
sin
v
+
1
t
sin
t/2 n
k
X
X
1
1
k
k−v q
(n + 1)
(1 + q)k v=0 v
k=0
sin t/2
cos v + 1 tcos t/2 + sin v + 1 t sin t/2 n
k
X
X k
1
1
q k−v k
(n + 1)
(1 + q) v=0 v
k=0
sin t/2
≤
n
k X
X
1
1
1
k
k−v
+O 1
O
q
(n + 1)
(1 + q)k v=0 v
t
k=0
#
"
#
"
n
n
k k X
X
X
X
1
1
1
1
k
k
q k−v + O
q k−v
O
(n + 1)t
(1 + q)k v=0 v
(n + 1)
(1 + q)k v=0 v
k=0
k=0
1
1
O
(n + 1) + O
(n + 1)
(n + 1)t
(n + 1)
1
O
+ O[1],
t
in view of |sin(v + 1)t| ≤ 1 for 0 ≤ t ≤ √πn ≤
This completes the proof of Lemma (4.2).
π
√
v
≤ t ≤ π.
5. Proof of Theorem 3.1
Let s̃n (x) denotes the partial sum of series (2), we have
Z π
cos n + 1/2 t
1
dt.
ψ(t)
sen (x) − fe(x) =
2π 0
sin t/2
Therefore, using (2) the (C,1)transform Cn1 of sen is given by
Z π
n cos n + 1/2 t
X
1
en1 − fe(x) =
dt.
C
ψ(t)
2π(n + 1) 0
k=0
sin t/2
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS)
47
^
1 q
Now denoting (C,^
1)(E, q) transform of sen as (C
n En ), we write
X
Z π
n
k 1
1
1
ψ(t) X k
k−v
^
1 E q ) − fe(x) =
tdt
(C
q
cos
v
+
n n
2π(n + 1)
(1 + q)k 0 sint 2t v=0 v
2
k=0
Z π
en (t) dt
=
ψ(t)G
0
=
Z
√
π/ n
+
0
=
Z
π
√
π/ n
I1 + I2 , (say).
We consider,
|I1 | ≤
Z
√
π/ n
0
en (t)dt
ψ(t)G
(21)
en (t) dt.
|ψ(t)|G
Using Lemma 4.1, Hölder’s inequality, Minkowski’s inequality and condition 18, we
have
r
1/r Z π/√n Z π/√n en (t)| s 1/s
t|ψ(t)|
ξ(t)|G
βr
sin t/2dt
dt
|I1 | ≤
ξ(t)
tsinβ t/2
0
0
√
Z π/ n en (t)| s 1/s
ξ(t)|G
1
dt
= O √
n
tsinβ t/2
0
Z π/√n s 1/s
ξ(t)(n + 1)t
ξ(t)
1
dt
+
= O √
n
t2 sinβ t/2
tsinβ t/2
0
√
s 1/s
s 1/s
Z π/ n Z π/√n √
1
ξ(t)
ξ(t)
dt
dt
= O √
+
O(
, as h → 0.
n)
t2+β
tβ
n
h
h
Since ξ(t) is a positive increasing function and using second mean value theorem
for integrals, we have
Z π/√n Z π/√n s 1/s
s 1/s
√
π
1
π
1
1
|I1 | = O √ ξ √
nξ √
dt
dt
+O
, as h → 0
2+β
n
n
t
n
tβ
h
h
Z π/√n
Z π/√n
1/s
1/s
√
1
1
1
−βs−2s
−βs
t
dt
t
dt
+O
, as h → 0.
nπξ √
= O √ πξ √
n
n
n
h
h
Note that
1
π
≤ πξ √ ,
ξ √
n
n
√
√
1
1
1
O √ ξ √ ( n)β+2−1/s + O ξ √ ( n)β+1−1/s
n
n
n
√
1
O ξ √ ( n)β+1−1/s
n
√
1 1
1
O ξ √ ( n)β+1/r ∴ + = 1, 1 ≤ r ≤ ∞.
n
r
s
|I1 | =
=
=
Now, we consider,
Rπ
e
√
|I2 | ≤ π/ n |ψ(t)|Gn (t) dt.
(22)
48
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
Using Lemma 4.2, Hölder’s inequality, Minkowski’s inequality and condition 19, we
have
|I2 | ≤
=
=
=
r 1/r Z π en (t)| s 1/s
t−δ sinβ t/2.|ψ(t)|
ξ(t)|G
dt
dt
√
√
ξ(t)
t−δ sinβ t/2
π/ n
π/ n
Z π en (t)| s 1/s
√
ξ(t)|G
O ( n)δ
dt
√
t−δ sinβ t/2
π/ n
Z π s 1/s
√
ξ(t)
ξ(t)
dt
+
O ( n)δ
√
t−δ+1 sinβ t/2 t−δ sinβ t/2
π/ n
Z π s 1/s Z π s 1/s √
ξ(t)
ξ(t)
O ( n)δ
dt
dt
.
+
√
√
t−δ+β+1
t−δ+β
π/ n
π/ n
Z
π
Now putting t = 1/y,
|I2 | ≤
√
O ( n)δ
Z
√
( n)/π
1/π
ξ(1/y)
y δ−β−1
s
dy
y2
1/s
+
Z
√
( n/π)
1/π
ξ(1/y)
y δ−β
s
dy
y2
1/s .
Since ξ(t) is a positive increasing function so ξ(1/y)
is also a positive increasing
1/y
function and using second mean value theorem for integrals, we have
|I2 | =
=
=
=
=
=
=
√
1/s Z (√n)/π
1/s #
ξ π/ n " Z (√n)/π
√ δ
dy
dy
√
.
+
O ( n)
π/ n
y −βs+δs+2
y −βs+δs+s+2
1/π
1/π
"
1/s 1/s #
√
√
√ δ+1
1
(
n)/π
(
n)/π
O ( n) ξ √
(y −δs−1+βs )1/π
.
+ (y −δs−1+βs−s )1/π
n
√
√
√
1
( n)−δ−1/s+β ) + ( n)−δ−1/s+β−1 .
O ( n)δ+1 ξ √
n
h
i
√
√
1
( n)−δ−1/s+β+δ+1 + ( n)−δ−1/s+β−1+δ+1 .
O ξ √
n
h
i 1 1
√
√
1
O ξ √
( n)β+1/r + ( n)−1+β+1/r ∴ + = 1, 1 ≤ r ≤ ∞
r
s
n
√
√
1
O ξ √ ( n)β+1/r 1 + ( n)−1
n
√
1
O ξ √ ( n)β+1/r .
(23)
n
Combining I1 and I2 yields
√ β+1/r
1
q
^
1
e
.
ξ √
|Cn En − f | = O ( n)
n
(24)
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS)
49
Now, using the Lr -norm of a function, we get
Z 2π
1/r
q
q
r
^
^
1
1
e
e
kCn En − f kr =
|Cn En − f | dx
0
r 1/r
√
1
( n)β+1/r ξ √
n
0
Z 2π 1/r √
1
= O ( n)β+1/r ξ √
dx
n
0
√
1
= O ( n)β+1/r ξ √
.
n
= O
Z
2π
This completes the proof of Theorem 3.1.
6. Applications
The theory of approximation is a very extensive field, which has various applications. As mentioned in [20], the Lp -space in general, and L2 and L∞ in particular
play an important role in the theory of signals and filters. From the point of view
of the applications, Sharper estimates of infinite matrices 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 ξ(t) = tα , 0 < α ≤ 1, then the weighted class W (Lr , ξ(t)), r ≥ 1
reduces to the class Lip(α, r), (1/r) < α < 1 and the degree of approximation of
a function fe(x), conjugate to a 2π - periodic function f belonging to the class
Lip(α, r), is given by
1
q
^
1
e
|Cn En − f kr = O √ α−1/r .
(25)
( n)
Proof. Putting β = 0, ξ(t) = tα , 0 < α ≤ 1 in Theorem 3.1, we have
Z 2π
1/r
^
^
1 E q − fek =
1 E q (f ; x) − fe(x)|r dx
kC
|
C
n
n
r
n
n
0
or,
or,
Z 2π
1/r
√
1
^
1 E q (f ; x) − fe(x)|r dx
=
O ( n)1/r ξ √
|C
n n
n
0
O(1) =
Z
0
2π
1/r 1
^
1 E q (f ; x) − fe(x)|r dx
|C
O √ 1/r 1 ,
n n
( n) ξ( √n )
since otherwise the right hand side of the above equation will not be O(1). Hence
α
√ 1/r
1
1
^
1 E q (f ; x) − fe(x) = O
C
√
= O √ α−1/r .
( n)
n n
n
( n
This completes the proof of Corollary 6.1
50
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
Corollary 6.2. If ξ(t) = tα , 0 < α ≤ 1 and r → ∞ in corollary 6.1, then f ∈ Lipα
and
α 1
^
1 E q − fe = O
C
√
.
(26)
n n
n
Proof. For r → ∞ in corollary 6.1, we get
α 1
^
^
1 E q − fek
1 E q (f ; x) − fe(x) = O
√
.
kC
=
sup
C
∞
n n
n n
n
0≤x≤2π
Thus, we have
^
1 E q (f ; x) − fe(x)
C
n n
≤
=
^
1 E q (f ; x) − fe(x)
C
n n
∞
q
^
1
e
sup Cn En (f ; x) − f (x)
x∈[0,2π]
=
−α √
O
n
This completes the proof of Corollary 6.2.
7. An Example
Since Hausdorff matrices commute, therefore it is very easy to find an example
of a positive non-convergent sequence which is (C, 1) summable. Let us consider
an infinite series
∞
X
(−1)n (2q + 1)n cos nx
n=0
The nth partial sums sn of series (27) at x =0 is given by
sn
=
n
X
(−1)r (2q + 1)r cos rx
r=0
≤
1 − (−1)n+1 (2q + 1)n+1
2(1 + q)
Since limn→∞ sn does not exist. Therefore the series (27) is non-convergent.
Now, the (E, q) transform of (27) is given by
n X
1
n
q
En =
q n−k sk , q > 0
q
(1 + q)n
k=0
n X
1 − (−1)k+1 (2q + 1)k+1
1
n
n−k
q
≤
k
(1 + q)n
2(1 + q)
k=0
=
=
1
(1 + 2q)(q − 1 − 2q)n
+
2(1 + q)
2(1 + q)(q + 1)n
1
(1 + 2q)(−1)n
+
2(1 + q)
2(1 + q)
(27)
DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS)
51
Here, limn→∞ Enq not exist. Hence the series (27) is not summable.
Now,
n
n 1 X
1 X 1 − (−1)k+1 (2q + 1)k+1
σn1 =
sk ≤
n+1
n+1
2(1 + q)
k=0
k=0
=
=
=
∞
X
(1 + 2q)
1
+
(−1)k (2q + 1)k
2(1 + q) 2(1 + q)(n + 1) n=0
1
(1 + 2q)
+
{1 + (2q + 1)n+1 }
2(1 + q) 4(1 + q)2 (n + 1)
1
(1 + 2q)
(1 + 2q)n+2
+
+
2
2(1 + q) 4(1 + q) (n + 1) 4(1 + q)2 (n + 1)
Here, limn→∞ σn1 does not exist, the series (27) is not (C, 1) summable.
In
n X
n
q n−k
k
k=0
Change the variable of summation by j = (n − k) to get
n 0 X
X
n
n
q j = (1 + q)n .
qj =
j
n−j
j=n
j=0
Now integrate both sides from 0 to x to get
j+1
x
n X
(1 + x)n+1 − 1
x
(1 + q)n+1
n
=
=
.
j
j+1
n+1
n+1
0
j=0
Thus
j
n X
(1 + x)n+1 − 1
x
n
=
.
j
j+1
x(n + 1)
j=0
Finally, the (C, 1) (E, q) transform of (27) is given by
n X
1
n
Enq Cn1 =
q n−k σk1
k
(1 + q)n
k=0
n X
(1 + 2q)
(1 + 2q)k+2
1
1
n
n−k
+
+
q
≤
k
(1 + q)n
2(1 + q) 4(1 + q)2 (k + 1) 4(1 + q)2 (k + 1)
k=0
n−k
n−k
n n X
(1 + 2q)
(1 + 2q)2 X n
q
1
q
n
+
−
+
=
n+2
n+2
k
k
2(1 + q) 4(1 + q)
(k + 1) 4(1 + q)
(k + 1)
k=0
k=0
2
n+1
n+1
1
(1 + 2q)
(1 + q)
−1
(1 + q)
−1
(1 + 2q)
=
+
+
2(1 + q) 4(1 + q)n+2
(n + 1)q
4(1 + q)n+2
(n + 1)q
1
1
1
(1 + 2q)2
(1 + 2q)
=
1−
1
−
+
.
+
2(1 + q) 4(1 + q)(n + 1)q
(1 + q)n+1
4(1 + q)(n + 1)q
(1 + q)n+1
1
Enq Cn1 → 2(1+q)
as n → ∞, the series (27) is (E, q)(C, 1) summable.
Therefore the series (27) is neither (E, q) summable nor (C, 1) summable but it
1
. Thus the product summability (E, q)(C, 1) is
is (E, q)(C, 1) summable to 2(1+q)
more powerful than (E, q) and (C, 1). Consequently (E, q)(C, 1) gives the better
52
V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA
approximation than the individual methods (E, q) and (C, 1).
Acknowledgement: Authors are grateful to the anonymous learned referees for
their valuable suggestions and comments, leading to an overall improvement in the
paper. Authors mention their sense of gratitude to Professor Hüseyin Bor, editorial
board member of BMAA and Professor Faton Z. Merovci, secretary of BMAA for
for their efforts in sending the reports of the manuscript timely. LNM computed
an example of (E, q)(C, 1) in its detail in the present manuscript.
Kejal Khatri and Lakshmi Narayan Mishra, one of the authors, are thankful to
the financial support of Ministry of Human Resource Development (MHRD), New
Delhi, Govt. of India, India to carry out their research article.
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Vishnu Narayan Mishra and Kejal Khatri
Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat,
Surat-395 007 (Gujarat), India.
E-mail address: vishnunarayanmishra@gmail.com; vishnu narayanmishra@yahoo.co.in and
kejal0909@gmail.com
Huzoor H. Khan
Department of Mathematics, Aligarh Muslim University
Aligarh, India.
E-mail address: huzoorkhan@yahoo.com
Lakshmi Narayan Mishra
L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opp. I.T.I., Ayodhya Main Road
Faizabad-224 001, (Uttar Pradesh).
Department of Mathematics, National Institute of Technology, Silchar - 788 010,
District - Cachar (Assam), India.
E-mail address: lakshminarayanmishra04@gmail.com; l n mishra@yahoo.co.in