# Degree of Approximation of Functionsf

Authors: Vishnu Narayan Mishra; Kejal Khatri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

A new estimate for the degree of approximation of a functionf
˜
∈
H
ω class by (
N
p
·
E
1
) means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001). Some corollaries have also been deduced from our main theorem.


---

## Body

## 1. Introduction

The degree of approximation of a functionf belonging to various classes using different summability method has been determined by several investigators like Khan [1, 2], V. N. Mishra and L. N. Mishra [3], Mishra et al. [4–6], and Mishra [7, 8]. Summability techniques were also applied on some engineering problems; for example, Chen and Jeng [9] 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 and Hong [10] used Cesàro sum regularization technique for hyper singularity of dual integral equation. Summability of Fourier series is useful for engineering analysis, for example, [11]. Recently, Mursaleen and Mohiuddine [12] discussed convergence methods for double sequences and their applications in various fields. In sequel, Alexits [13] studied the degree of approximation of the functions in H
α class by the Cesàro means of their Fourier series in the sup-norm. Chandra ([14, 15]), Mohapatra and Chandra ([16, 17]), and Szal [18] have studied the approximation of functions in Hölder metric. Mishra et al. [6] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals. In 2008, Singh and Mahajan [19] studied error bound of periodic signals in the Hölder metric and generalized the result of Lal and Yadav [20] under much more general assumptions. Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. Especially, Psarakis and Moustakides [21] presented a new L
2 based method for designing the finite impulse response (FIR) digital filters and got corresponding optimum approximations having improved performance. We also discuss an example when the Fourier series of the signal has Gibbs phenomenon.For a2
π-periodic signal f
∈
L
p
:
=
L
p
[
0
,
2
π
]
,
p
≥
1, periodic integrable in the sense of Lebesgue. Then the Fourier series of f
(
x
) is given by
(1)
f
(
x
)
~
a
0
2
+
∑
n
=
1
∞
(
a
n
cos
⁡
n
x
+
b
n
sin
n
x
)
≡
∑
n
=
0
∞
A
n
(
x
)
,
with (
n
+
1
)th partial sum s
n
(
f
;
x
) called trigonometric polynomial of degree (or order) n and given by
(2)
s
n
(
f
;
x
)
:
=
a
0
2
+
∑
k
=
1
n
(
a
k
cos
⁡
k
x
+
b
k
sin
k
x
)
,
n
∈
ℕ
with
s
0
(
f
;
x
)
=
a
0
2
.
The conjugate series of Fourier series (1) is given by
(3)
∑
n
=
1
∞
(
b
n
cos
⁡
n
x
-
a
n
sin
n
x
)
≡
∑
n
=
1
∞
B
n
(
x
)
,
with nth partial sum s
~
n
(
f
;
x
).Letω
(
t
) and ω
*
(
t
) denote two given moduli of continuity such that
(4)
(
ω
(
t
)
)
β
/
α
=
O
(
ω
*
(
t
)
)
as
t
⟶
0
+
,
s
s
s
d
s
s
for
0
≤
β
<
α
≤
1
.
Let C
2
π denote the Banach space of all 2
π-periodic continuous functions defined on [
-
π
,
π
] under the sup-norm. The space L
p
[
0,2
π
] where p
=
∞ includes the space C
2
π. For some positive constant K, the function space H
ω is defined by
(5)
H
ω
=
{
f
∈
C
2
π
:
|
f
(
x
)
-
f
(
y
)
|
≤
K
ω
(
|
x
-
y
|
)
}
,
with norm ∥
·
∥
ω
* defined by
(6)
∥
f
∥
ω
*
=
∥
f
∥
c
+
sup
⁡
x
,
y
[
Δ
ω
*
f
(
x
,
y
)
]
,
where ω
(
t
) and ω
*
(
t
) are increasing functions of t,
(7)
∥
f
∥
c
=
sup
⁡
0
≤
x
≤
2
π
|
f
(
x
)
|
,
(8)
Δ
ω
*
f
(
x
,
y
)
=
|
f
(
x
)
-
f
(
y
)
|
ω
*
(
|
x
-
y
|
)
,
x
≠
y
,
with the understanding that Δ
0
f
(
x
,
y
)
=
0. If there exist positive constants B and K such that ω
(
|
x
-
y
|
)
≤
B
|
x
-
y
|
α and ω
*
(
|
x
-
y
|
)
≤
K
|
x
-
y
|
β
,
0
≤
β
<
α
≤
1, then the space
(9)
H
α
=
{
f
∈
C
2
π
:
|
f
(
x
)
-
f
(
y
)
|
≤
K
|
x
-
y
|
α
,
0
<
α
≤
1
}
is Banach space [22] and the metric induced by the norm ∥
·
∥
α on H
α is said to be Hölder metric. Clearly H
α is a Banach space which decreases as α increases; that is,
(10)
H
α
⊆
H
β
⊆
C
2
π
,
for
0
≤
β
<
α
≤
1
.
Let ∑
n
=
0
∞
a
n be a given infinite series with the sequence of nth partial sums {
s
n
}. Let {
p
n
} be a nonnegative sequence of constants, real or complex, and let us write
(11)
P
n
=
∑
k
=
0
n
p
k
≠
0
∀
n
≥
0
,
p
-
1
=
0
=
P
-
1
,
P
n
⟶
∞
as
n
⟶
∞
.
The sequence to sequence transformation
(12)
t
n
N
(
f
;
x
)
=
1
P
n
∑
k
=
0
n
p
n
-
k
s
k
(
f
;
x
)
defines the sequence {
t
n
N
} of Nörlund means of the sequence {
s
n
}, generated by the sequence of coefficients {
p
n
}. The series ∑
n
=
0
∞
a
n is said to be summable N
p to the sum s if lim
⁡
n
→
∞
t
n
N exists and is equal to s.In the special case in which(13)
p
n
=
(
n
+
α
-
1
α
-
1
)
=
Γ
(
n
+
α
)
Γ
(
n
+
1
)
Γ
(
α
)
;
(
α
>
0
)
,
then Nörlund summability N
p reduces to the familiar (
C
,
α
) summability.An infinite series∑
n
=
0
∞
a
n is said to be (
C
,
1
) summable to s if
(14)
(
C
,
1
)
=
1
(
n
+
1
)
∑
k
=
0
n
s
k
⟶
s
as
n
⟶
∞
.
The E
1 transform is defined as the nth partial sum of E
1 summability and we denote it by E
~
n
1.If(15)
E
~
n
1
(
f
)
=
E
~
n
1
(
f
;
x
)
=
1
2
n
∑
k
=
0
n
(
n
k
)
s
~
k
⟶
s
,
as
n
⟶
∞
,
then the infinite series ∑
n
=
0
∞
a
n is said to be (
E
,
1
) summable to s.The(
C
,
1
) transform of the (
E
,
1
) transform E
n
1 defines the (
C
,
1
)
(
E
,
1
) transform of the partial sums s
n of the series ∑
n
=
0
∞
a
n; that is, the product summability (
C
,
1
)
(
E
,
1
) is obtained by superimposing (
C
,
1
) summability on (
E
,
1
) summability. Thus, if
(16)
(
C
E
)
n
1
=
1
(
n
+
1
)
∑
k
=
0
n
E
k
1
=
1
(
n
+
1
)
∑
k
=
0
n
1
2
k
∑
r
=
0
k
(
k
r
)
s
r
⟶
s
,
as
n
⟶
∞
,
where E
n
1 denotes the (
E
,
1
) transform of s
n, then the series ∑
n
=
0
∞
a
n with the partial sums s
n is said to be summable (
C
,
1
)
(
E
,
1
) to the definite number s and we can write
(17)
(
C
E
)
n
1
⟶
s
[
(
C
,
1
)
(
E
,
1
)
]
,
as
n
⟶
∞
.
The N
p transform of the E
1 transform defines (
N
p
·
E
1
) product transform and denote it by t
~
n
N
E
(
f
). If
(18)
t
~
n
N
E
(
f
)
=
t
~
n
N
E
(
f
;
x
)
=
1
P
n
∑
k
=
0
n
p
n
-
k
E
~
k
1
(
f
)
⟶
s
,
as
n
⟶
∞
,
then the infinite series ∑
n
=
0
∞
a
n is said to be (
N
p
·
E
1
) summable to s.We note thatE
n
1
,
(
C
E
)
n
1, and t
~
n
N
E are also trigonometric polynomials of degree (or order) n.The relation between Cesàro mean and Fejér mean is given by(19)
σ
n
(
x
)
=
1
2
π
∫
0
2
π
[
f
(
x
+
t
)
+
f
(
x
-
t
)
]
F
n
(
t
)
d
t
,
where the Fejér mean is
(20)
F
0
(
t
)
=
1
,
F
n
(
t
)
=
{
1
n
+
1
(
sin
(
n
+
1
)
(
t
/
2
)
sin
(
t
/
2
)
)
2
,
t
∉
2
π
Z
,
n
≥
1
(
n
+
1
)
,
t
∈
2
π
Z
,
and Cesàro mean is
(21)
σ
n
=
s
0
(
x
)
+
s
1
(
x
)
+
⋯
+
s
n
-
1
(
x
)
n
=
1
n
∑
k
=
0
n
-
1
s
k
(
x
)
.
Some important particular cases are as follows:(1)
(
H
,
1
/
(
n
+
1
)
)
·
(
E
1
) means, when a
n
,
k
=
1
/
(
n
-
k
+
1
)
log
⁡
n.
(2)
(
N
,
p
n
,
q
n
)
·
E
1 means, a
n
,
k
=
p
n
-
k
q
k
/
R
n, where R
n
=
∑
k
=
0
n
p
k
q
n
-
k
≠
0
.Remark 1.
If we takeω
(
|
x
-
y
|
)
=
|
x
-
y
|
α, then H
ω reduces to H
α class.The conjugate functionf
~
(
x
) is defined for almost every x by
(22)
f
~
(
x
)
=
-
1
2
π
∫
0
π
ψ
(
t
)
cot
t
2
d
t
=
lim
⁡
h
→
0
⁡
(
-
1
2
π
∫
h
π
ψ
(
t
)
cot
t
2
d
t
)
;
see [23, page 131].We write throughout the paper(23)
ψ
x
(
t
)
=
ψ
(
t
)
=
f
(
x
+
t
)
-
f
(
x
-
t
)
,
ϕ
x
(
t
)
=
f
(
x
+
t
)
-
2
f
(
x
)
+
f
(
x
-
t
)
,
(
N
p
·
E
1
)
n
(
t
)
=
1
P
n
∑
k
=
0
n
p
n
-
k
cos
⁡
k
(
t
/
2
)
cos
⁡
(
k
+
1
)
(
t
/
2
)
sin
t
/
2
.
We note that the series, conjugate to a Fourier series, is not necessarily a Fourier series [23, 24]. Hence, a separate study of conjugate series is desirable and attracted the attention of researchers.


## 2. Known Results

In 2001, Lal and Yadav [20] established the following theorem to estimate the error between the input signal f
(
x
) and the signal obtained after passing through the (
C
,
1
)
(
E
,
1
)- transform.Theorem 2 (see [20]).
If a functionf
:
R
→
R is 2
π-periodic function and belongs to class Lip α, 0
<
α
≤
1, then degree of approximation by (
C
,
1
)
(
E
,
1
) means of its Fourier series is given by
(24)
∥
t
n
C
1
·
E
1
(
f
)
-
f
(
x
)
∥
∞
=
{
O
(
n
-
α
)
,
0
<
α
<
1
O
(
log
⁡
n
n
)
,
α
=
1
.
Recently, Singh and Mahajan [19] generalized the above result under more general assumptions. They proved the following.Theorem 3 (see [19]).
Letω
(
t
) defined in (5) be such that
(25)
∫
t
π
ω
(
u
)
u
2
d
u
=
O
(
H
(
t
)
)
,
H
(
t
)
≥
0
,
∫
0
t
H
(
u
)
d
u
=
O
(
t
H
(
t
)
)
,
a
s
t
⟶
0
+
;
then, for 0
<
β
≤
α
≤
1 and f
∈
H
ω, we have
(26)
∥
t
n
C
1
·
E
1
(
f
)
-
f
(
x
)
∥
ω
*
=
O
(
(
(
n
+
1
)
-
1
H
(
π
n
+
1
)
)
1
-
β
/
α
)
.Theorem 4 (see [19]).
Considerω
(
t
) defined in (5) and for 0
<
β
≤
α
≤
1 and f
∈
H
ω, we have
(27)
∥
t
n
C
1
·
E
1
(
f
)
-
f
(
x
)
∥
ω
*
=
O
(

(
ω
(
π
n
+
1
)

)
1
-
β
/
α

+

(

(
n
+
1
)
-
1
∑
k
=
1
n
+
1
ω
(
1
k
+
1
)

)
1
-
β
/
α
)
.


## 3. Main Theorem

In this paper, we prove a theorem on the degree of approximation of a functionf
~
(
x
), conjugate to a 2
π-periodic function f belonging to f
~
∈
H
ω class by (
N
p
·
E
1
) means of conjugate series of its Fourier series. This work generalizes the results of Singh and Mahajan [19] on (
N
p
·
E
1
) summability of conjugate Fourier series. We will measure the error between the input signal f
~
(
x
) and the processed output signal t
~
n
N
E
(
f
;
x
)
=
(
1
/
P
n
)
∑
k
=
0
n
p
n
-
k
E
~
k
1
(
f
), by establishing the following theorems.Theorem 5.
The functionsω satisfy the following conditions:
(28)
∫
t
π
ω
(
u
)
u
2
d
u
=
O
(
H
(
t
)
)
,
H
(
t
)
≥
0
,
(29)
∫
0
t
H
(
u
)
d
u
=
O
(
t
H
(
t
)
)
,
a
s
t
⟶
0
+
,
where ω
(
t
) and ω
*
(
t
) are increasing functions of t. Let N
p be the Nörlund summability matrix generated by the nonnegative {
p
n
} such that
(30)
(
n
+
1
)
p
n
=
O
(
P
n
)
,
∀
n
≥
0
.
Then, for f
~
∈
H
ω
,
0
<
β
≤
α
≤
1, we have
(31)
∥
t
~
n
N
E
(
f
)
-
f
~
(
x
)
∥
ω
*
=
O
{
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
(
log
⁡
(
n
+
1
)
)
β
/
α
f
f
f
f
f
×
[
(
n
+
1
)
-
1
H
(
π
n
+
1
)
]
1
-
β
/
α
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
}
,
and if ω
(
t
) satisfies (28), then for f
~
∈
H
ω
,
0
<
β
≤
α
≤
1, we have
(32)
∥
t
~
n
N
E
(
f
)
-
f
~
(
x
)
∥
ω
*
=
O
{
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
(
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
log
⁡
(
n
+
1
)
[
ω
(
π
n
+
1
)
]
1
-
β
/
α
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
+
[

(
1
n
+
1
)

∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α

)

}
.Remark 6.
The product transform(
N
p
·
E
1
) plays an important role in signal theory as a double digital filter and theory of Machines in Mechanical Engineering [4, 5].


## 4. Lemmas

In order to prove our main Theorem5, we require the following lemmas.Lemma 7.
Ifψ
x
(
t
)
=
ψ
(
t
)
=
f
(
x
+
t
)
-
f
(
x
-
t
), then for f
~
∈
H
ω, we have
(33)
|
ψ
x
(
t
)
-
ψ
y
(
t
)
|
≤
2
M
ω
(
|
x
-
y
|
)
,
(34)
|
ψ
x
(
t
)
-
ψ
y
(
t
)
|
≤
2
M
ω
(
|
t
|
)
.
It is easy to verify the following.Lemma 8.
Let{
p
n
} be a nonnegative and nonincreasing sequence satisfies (30); then for 0
<
t
≤
π
/
(
n
+
1
), we have
(35)
(
N
p
·
E
1
)
n
(
t
)
=
O
(
1
t
)
.Proof.
Consider(36)
(
N
p
·
E
1
)
n
(
t
)
=
1
P
n
∑
k
=
0
n
p
n
-
k
cos
⁡
k
(
t
/
2
)
cos
⁡
(
k
+
1
)
(
t
/
2
)
sin
t
/
2
.
Using condition (30), sin
n
t
≤
n
t
,
sin
(
t
/
2
)
≥
(
t
/
π
), for 0
<
t
≤
π and |
cos
⁡
k
(
t
/
2
)
|
≤
1, we have
(37)
(
N
p
·
E
1
)
n
(
t
)
≤
p
n
P
n
∑
k
=
0
n
cos
⁡
k
(
t
/
2
)
cos
⁡
(
k
+
1
)
(
t
/
2
)
sin
t
/
2
=
O
(
1
t
(
n
+
1
)
)
[
∑
k
=
0
n
cos
⁡
k
(
t
2
)
cos
⁡
(
k
+
1
)
(
t
2
)
]
=
(
1
t
(
n
+
1
)
)
∑
k
=
0
n
Re
{
cos
⁡
k
(
t
2
)
e
i
(
k
+
1
)
t
/
2
}
=
(
1
t
(
n
+
1
)
)
Re
{
e
i
t
/
2
(
1
-
cos
⁡
n
+
1
(
t
/
2
)
e
i
(
n
+
1
)
t
/
2
1
-
cos
⁡
(
t
/
2
)
e
i
t
/
2
)
}
=
(
1
t
2
(
n
+
1
)
)
Re
{
i
-
i
cos
⁡
(
n
+
1
)
(
t
/
2
)
cos
⁡
n
+
1
(
t
/
2
)
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
+
sin
(
n
+
1
)
(
t
/
2
)
cos
⁡
n
+
1
(
t
/
2
)
}
=
O
(
1
t
2
(
n
+
1
)
)
sin
(
n
+
1
)
(
t
/
2
)
cos
⁡
n
+
1
(
t
/
2
)
.
=
O
(
1
t
2
(
n
+
1
)
(
n
+
1
)
t
)
=
O
(
1
t
)
.Lemma 9.
Let{
p
n
} be a nonnegative and nonincreasing sequence satisfies (30); then for
π
/
(
n
+
1
)
≤
t
≤
π, we get
(38)
(
N
p
·
E
1
)
n
(
t
)
=
O
(
1
t
2
(
n
+
1
)
)
.Proof.
Consider(39)
(
N
p
·
E
1
)
n
(
t
)
=
1
P
n
∑
k
=
0
n
p
n
-
k
cos
⁡
k
(
t
/
2
)
cos
⁡
(
k
+
1
)
(
t
/
2
)
sin
t
/
2
.
Using condition (30), sin
(
t
/
2
)
≥
(
t
/
π
)
,
for
0
<
t
≤
π
,
|
sin
t
|
≤
1 for all t and |
cos
⁡
k
(
t
/
2
)
|
≤
1, we have
(40)
(
N
p
·
E
1
)
n
(
t
)
≤
p
n
P
n
∑
k
=
0
n
cos
⁡
k
(
t
/
2
)
cos
⁡
(
k
+
1
)
(
t
/
2
)
sin
t
/
2
.
=
(
1
t
(
n
+
1
)
)
∑
k
=
0
n
Re
{
cos
⁡
k
(
t
2
)
e
i
(
k
+
1
)
t
/
2
}
=
(
1
t
(
n
+
1
)
)
Re
{
e
i
t
/
2
(
1
-
cos
⁡
n
+
1
(
t
/
2
)
e
i
(
n
+
1
)
t
/
2
1
-
cos
⁡
(
t
/
2
)
e
i
t
/
2
)
}
=
O
(
1
t
2
(
n
+
1
)
)
sin
(
n
+
1
)
(
t
/
2
)
cos
⁡
n
+
1
(
t
/
2
)
.
=
O
(
1
t
2
(
n
+
1
)
)
.Lemma 10 (see [19]).
Ifω
(
t
) satisfies conditions (28) and (29), then
(41)
∫
0
u
t
-
1
ω
(
t
)
d
t
=
O
(
u
H
(
u
)
)
,
as
u
⟶
0
+
.


## 5. Proof of Theorem5

The integral representationofs
~
n
(
f
;
x
) is given by
(42)
s
~
n
(
f
;
x
)
=
∑
n
=
1
∞
(
b
n
cos
⁡
n
x
-
a
n
sin
n
x
)
=
-
1
π
∫
0
π
ψ
x
(
t
)
(
cot
(
t
/
2
)
cos
⁡
(
n
+
1
/
2
)
t
2
sin
(
t
/
2
)
)
d
t
and, therefore,
(43)
s
~
n
(
f
;
x
)
-
f
~
(
x
)
=
1
2
π
∫
0
π
ψ
x
(
t
)
cos
⁡
(
n
+
1
/
2
)
t
sin
t
/
2
d
t
.
Denoting E
1 means of s
~
n
(
f
;
x
) by E
~
n
1
(
x
), we have
(44)
1
2
n
∑
k
=
0
n
(
n
k
)
{
s
~
n
(
f
;
x
)
-
f
~
(
x
)
}
=
1
2
(
n
+
1
)
π
∫
0
π
ψ
x
(
t
)
sin
t
/
2
∑
k
=
0
n
(
n
k
)
cos
⁡
(
k
+
1
/
2
)
t
d
t
,
E
~
n
1
(
x
)
-
f
~
(
x
)
=
1
2
(
n
+
1
)
π
∫
0
π
ψ
x
(
t
)
sin
t
/
2
∑
k
=
0
n
(
n
k
)
cos
⁡
(
k
+
1
/
2
)
t
d
t
=
1
2
(
n
+
1
)
π
∫
0
π
ψ
x
(
t
)
sin
t
/
2
Re
{
∑
k
=
0
n
(
n
k
)
e
i
(
k
+
1
/
2
)
t
}
d
t
=
1
2
(
n
+
1
)
π
∫
0
π
ψ
x
(
t
)
sin
t
/
2
Re
{
e
i
t
/
2
(
1
+
e
i
t
)
n
}
d
t
=
1
2
(
n
+
1
)
π
∫
0
π
ψ
x
(
t
)
sin
t
/
2
Re
{
2
n
cos
⁡
n
(
t
2
)
e
i
(
n
+
1
)
t
/
2
}
d
t
=
1
2
π
∫
0
π
ψ
x
(
t
)
cos
⁡
n
(
t
/
2
)
cos
⁡
(
n
+
1
)
(
t
/
2
)
sin
t
/
2
d
t
.
Now, using condition (30), then N
p transform of E
1 transform is given by
(45)
|
t
~
n
N
E
(
f
)
-
f
~
(
x
)
|
=
1
2
π
∫
0
π
|
ψ
x
(
t
)
|
t
d
d
d
d
d
d
×
|
∑
k
=
0
n
p
n
-
k
P
n
cos
⁡
k
(
t
2
)
cos
⁡
(
k
+
1
)
(
t
2
)
|
d
t
=
O
(
1
n
+
1
)
∫
0
π
|
ψ
x
(
t
)
|
t
d
d
d
d
d
d
d
d
d
d
d
×
|
∑
k
=
0
n
cos
⁡
k
(
t
2
)
cos
⁡
(
k
+
1
)
(
t
2
)
|
d
t
.
Set
(46)
E
n
(
x
)
=
|
t
~
n
N
E
(
f
)
-
f
~
(
x
)
|
=
O
(
1
n
+
1
)
×
∫
0
π
|
ψ
x
(
t
)
|
t
|
∑
k
=
0
n
cos
⁡
k
(
t
2
)
cos
⁡
(
k
+
1
)
(
t
2
)
|
d
t
,
E
n
(
x
,
y
)
=
|
E
n
(
x
)
-
E
n
(
y
)
|
=
O
(
1
n
+
1
)
∫
0
π
|
ψ
x
(
t
)
-
ψ
y
(
t
)
|
t
d
d
d
d
d
d
d
d
d
d
d
d
×
|
∑
k
=
0
n
cos
⁡
k
(
t
2
)
cos
⁡
(
k
+
1
)
(
t
2
)
|
d
t
=
O
(
1
n
+
1
)
[
∫
0
π
/
(
n
+
1
)
+
∫
π
/
(
n
+
1
)
π
]
=
I
1
+
I
2
,
(
say
)
.
Now, using (34) and Lemma 10, assume that ω
(
t
) satisfies (28) and (29); we get
(47)
I
1
=
O
(
1
)
∫
0
π
/
(
n
+
1
)
t
-
1
ω
(
t
)
d
t
=
O
(
(
n
+
1
)
-
1
H
(
π
n
+
1
)
)
.
Now, using (34) and Lemma 10, assume that ω
(
t
) satisfies (28) and (29); we get
(48)
I
2
=
O
(
1
n
+
1
)
∫
π
/
(
n
+
1
)
π
t
-
2
ω
(
t
)
d
t
=
O
(
(
n
+
1
)
-
1
H
(
π
n
+
1
)
)
.
Now, using (33), Lemma 8, we get
(49)
I
1
=
O
(
1
n
+
1
)
∫
0
π
/
(
n
+
1
)
|
ψ
x
(
t
)
-
ψ
y
(
t
)
|
t
d
t
=
O
(
1
n
+
1
)
∫
0
π
/
(
n
+
1
)
ω
(
|
x
-
y
|
)
t
d
t
=
O
(
ω
(
|
x
-
y
|
)
)
{
∫
0
π
/
(
n
+
1
)
1
t
d
t
}
,
where
0
<
t
≤
π
(
n
+
1
)
=
O
(
log
⁡
(
n
+
1
)
ω
(
|
x
-
y
|
)
)
.
Now, using (33), Lemma 9, we get
(50)
I
2
=
O
(
1
n
+
1
)
∫
π
/
(
n
+
1
)
π
t
-
2
ω
(
|
x
-
y
|
)
d
t
=
O
(
1
n
+
1
ω
(
|
x
-
y
|
)
)
∫
π
/
(
n
+
1
)
π
t
-
2
d
t
=
O
(
ω
(
|
x
-
y
|
)
)
.
Using the fact that we may write I
k
=
I
k
1
-
β
/
α
I
k
β
/
α
,
k
=
1,2, and combining (47) and (49), we get
(51)
I
1
=
O
(
[
(
n
+
1
)
-
1
H
(
π
n
+
1
)
]
1
-
β
/
α
d
f
d
d
×
[
log
⁡
(
n
+
1
)
ω
(
|
x
-
y
|
)
]
β
/
α
[
(
n
+
1
)
-
1
H
(
π
n
+
1
)
]
1
-
β
/
α
)
.
Combining (48) and (50), we get
(52)
I
2
=
O
(
[
(
n
+
1
)
-
1
H
(
π
n
+
1
)
]
1
-
β
/
α
[
ω
(
|
x
-
y
|
)
]
β
/
α
)
.
Therefore, from (8), (51), and (52), we have
(53)
sup
⁡
x
,
y
|
Δ
ω
*
E
(
x
,
y
)
|
=
sup
⁡
x
,
y
|
E
n
(
x
)
-
E
n
(
y
)
|
ω
*
(
|
x
-
y
|
)
=
O
{
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
(
log
⁡
(
n
+
1
)
)
β
/
α
d
d
d
d
d
×
[
(
n
+
1
)
-
1
H
(
π
n
+
1
)
]
1
-
β
/
α
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
}
.
Since
(54)
∥
E
n
(
x
)
∥
c
=
sup
⁡
0
≤
x
≤
2
π
|
t
~
n
T
·
E
1
(
f
)
-
f
~
(
x
)
|
,
it follows from (47) and (48) that
(55)
∥
E
n
(
x
)
∥
c
=
O
(
(
n
+
1
)
-
1
H
(
π
n
+
1
)
)
.
Combining (53) and (55), we get
(56)
∥
t
~
n
N
E
(
f
)
-
f
~
(
x
)
∥
ω
*
=
O
{
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
(
log
⁡
(
n
+
1
)
)
β
/
α
f
f
f
f
f
f
×
[
(
n
+
1
)
-
1
H
(
π
n
+
1
)
]
1
-
β
/
α
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
}
.
To prove (32), if ω
(
t
) satisfies only (28), then, using (34) and second mean value theorem for integrals, we have
(57)
I
1
=
O
(
ω
(
π
/
(
n
+
1
)
)
)
{
∫
0
π
/
(
n
+
1
)
1
t
d
t
}
,
where
0
<
t
≤
π
(
n
+
1
)
=
O
(
log
⁡
(
n
+
1
)
ω
(
π
n
+
1
)
)
.
Combining (49) and (57), we get
(58)
I
1
=
O
(
[
log
⁡
(
n
+
1
)
ω
(
π
n
+
1
)
]
1
-
β
/
α
d
d
d
d
×
[
log
⁡
(
n
+
1
)
ω
(
|
x
-
y
|
)
]
β
/
α
[
log
⁡
(
n
+
1
)
ω
(
π
n
+
1
)
]
1
-
β
/
α
)
.
Again, if ω
(
t
) satisfies only (28), then using (34) and second mean value theorem for integrals, we have
(59)
I
2
=
O
(
1
n
+
1
)
∫
π
/
(
n
+
1
)
π
ω
(
t
)
t
2
d
t
=
O
(
1
n
+
1
)
∫
1
n
+
1
ω
(
π
t
)
d
t
=
O
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
.
Combining (50) and (59), we get
(60)
I
2
=
O
(
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
[
ω
(
|
x
-
y
|
)
]
β
/
α
)
.
Therefore,
(61)
sup
⁡
x
,
y
|
Δ
ω
*
E
(
x
,
y
)
|
=
O
{
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
d
d
d
d
d
×
(
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
log
⁡
(
n
+
1
)
[
ω
(
π
n
+
1
)
]
1
-
β
/
α
d
d
d
d
d
d
d
d
d
+
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
)
}
,
(62)
∥
E
n
(
x
)
∥
c
=
O
(
log
⁡
(
n
+
1
)
ω
(
π
n
+
1
)
)
dddddddd
+
O
(
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
)
.
Combining (61) and (62), we get
(63)
∥
t
~
n
N
E
(
f
)
-
f
~
(
x
)
∥
ω
*
=
O
{
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
ω
(
|
x
-
y
|
)
β
/
α
ω
*
(
|
x
-
y
|
)
d
d
d
d
d
d
×
(
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α

log
⁡
(
n
+
1
)
[
ω
(
π
n
+
1
)
]
1
-
β
/
α
d
d
d
d
d
d
g
d
d
d
+
[
(
1
n
+
1
)
∑
k
=
0
n
ω
(
π
n
+
1
)
]
1
-
β
/
α
)
}
.
This completes the proof of Theorem 5.


## 6. Applications

The theory of approximation is a very extensive field and the study of theory of trigonometric approximation is of great mathematical interest and of great practical importance. As mentioned in [21], the L
p space in general and L
2 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 [25] are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix valued functions and enable to investigate perturbations of matrix valued functions and compare them. The following corollaries may be derived from Theorem 5.Ifω
(
|
x
-
y
|
)
≤
B
|
x
-
y
|
α
,
ω
*
(
|
x
-
y
|
)
≤
K
|
x
-
y
|
β
,
0
≤
β
<
α
≤
1 and set
(64)
H
(
u
)
=
{
u
α
-
1
,
0
<
α
<
1
log
⁡
(
1
u
)
,
α
=
1
,
then we get Corollary 11.Corollary 11.
Iff
~
∈
H
α
,
0
<
β
≤
α
≤
1, then
(65)
∥
t
~
n
N
E
(
f
)
-
f
~
(
x
)
∥
β
=
{
O
(
(
log
⁡
(
n
+
1
)
)
β
/
α
(
n
+
1
)
β
-
α
)
,
0
<
α
<
1
O
(
log
⁡
(
n
+
1
)
(
n
+
1
)
β
-
1
)
,
α
=
1
.
If we put β
=
0 in Corollary 11, then we get Corollary 12.Corollary 12.
Iff
~
∈  Lip α
,
0
<
α
≤
1, then
(66)
∥
t
~
n
N
E
(
f
)
-
f
~
(
x
)
∥
c
=
{
O
(
(
n
+
1
)
-
α
)
,
0
<
α
<
1
O
(
log
⁡
(
n
+
1
)
(
n
+
1
)
)
,
α
=
1
.


## 7. Example

In the example, we see how the sequence of averages (i.e.,σ
n
1
(
x
)-means or (
C
,
1
) mean) and Nörlund mean N
p of partial sums of a Fourier series is better behaved than the sequence of partial sums s
n
(
x
) itself.Let(67)
f
(
x
)
=
{
-
1
,
-
π
≤
x
<
0
,
1
,
0
≤
x
<
π
,
with f
(
x
+
2
π
)
=
f
(
x
) for all real x. Fourier series of f
(
x
) is given by
(68)
2
π
∑
n
=
1
∞
1
-
(
-
1
)
n
n
sin
n
x
,
-
π
≤
x
≤
π
.
Then nth partial sum s
n
(
x
) of Fourier series (68) and nth Cesàro sum for δ
=
1, that is, σ
n
1
(
x
) for the series (68), are given by
(69)
s
n
(
x
)
=
4
π
(
sin
x
+
1
3
sin
3
x
+
⋯
+
1
n
sin
n
x
)
,
σ
n
1
(
x
)
=
2
π
∑
k
=
1
n
(
1
-
k
n
)
(
1
-
(
-
1
)
k
k
)
sin
k
x
.
From Theorem 20 of Hardy’s “Divergent Series,” if a Nörlund method N
p has increasing weights {
p
n
}, then it is stronger than (
C
,
1
).Now, takeN
p to be the Nörlund matrix generated by p
n
=
n
+
1, then Nörlund means N
p is given by
(70)
t
n
N
(
f
;
x
)
=
2
(
n
+
1
)
(
n
+
2
)
∑
k
=
0
n
(
n
-
k
+
1
)
s
k
(
f
;
x
)
.
In this graph, we observe that σ
n
1
(
x
)
,
t
n
N
(
f
;
x
) converges to f
(
x
) faster than s
n
(
x
) in the interval [
-
π
,
π
]. We further note that near the points of discontinuities, that is, -
π, 0 and π, the graph of s
5 and s
10 shows peaks that move closer to the line passing through points of discontinuity as n increases (Gibbs phenomenon), but in the graph of σ
n
1
(
x
)
,
t
n
N
(
f
;
x
)
,
n
=
5
,
10 the peaks become flatter (Figure 1). The Gibbs phenomenon is an overshoot a peculiarity of the Fourier series and other eigen function series at a simple discontinuity; that is, the convergence of Fourier series is very slow near the point of discontinuity. Thus, the product summability means of the Fourier series of f
(
x
) overshoot the Gibbs Phenomenon and show the smoothing effect of the method. Thus, σ
n
1
(
x
)
,
t
n
N
(
f
;
x
) is the better approximant than s
n
(
x
) and N
p method is stronger than (C
,
1) method.Graph off
(
x
) (blue), s
n
(
x
) (pink), σ
n
1
(
x
) (yellow), t
n
N
(
f
;
x
) (green), n
=
5 and 10.
(a)
(b)


## 8. Conclusion

Several results concerning the degree of approximation of periodic signals (functions) by product summability means of Fourier series and conjugate Fourier series in generalized Hölder metric and Hölder metric have been reviewed. Using graphical representationσ
n
1
(
x
)
,
t
n
N
(
f
;
x
) is a better approximant to s
n
(
x
), but till now nothing has been done to show this. Some interesting application of the operator (
N
p
·
E
1
) used in this paper is pointed out in Remark 6. Also, the result of our theorem is more general rather than the results of any other previous proved theorems, which will enrich the literate of summability theory of infinite series.



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*Source: 837408-2014-04-30.xml*