# An Estimate of the Rate of Convergence of the Fourier Series in the Generalized Hölder Metric by Delayed Arithmetic Mean

Authors: L. Nayak; G. Das; B. K. Ray
Journal: International Journal of Analysis
 (2014)

---

## Abstract

We study the rate of convergence problem of the Fourier series by Delayed Arithmetic Mean in the generalized Hölder metric(
H
p
(
w
)
) space which was earlier introduced by Das, Nath, and Ray and obtain a sharper estimate of Jackson's order.


---

## Body

## 1. Definition

Letf be a 2
π-periodic function such that f
∈
L
p
[
0,2
π
]
,
p
≥
1. Let the Fourier series of f at t
=
x be given by
(1)
1
2
a
0
+
∑
n
=
1
∞
‍
(
a
n
cos
⁡
n
x
+
b
n
sin
n
x
)
.
Let
(2)
ϕ
x
(
t
)
=
1
2
{
f
(
x
+
t
)
+
f
(
x
-
t
)
-
2
f
(
x
)
}
.
Let S
n
(
x
) be the nth partial sum of (1). Then it is known ([1], page 50) that
(3)
S
n
(
x
)
-
f
(
x
)
=
2
π
∫
0
π
‍
ϕ
x
(
t
)
D
n
(
t
)
d
t
,
where D
n
(
t
)
=
sin
(
n
+
1
/
2
)
t
/
2
sin
(
t
/
2
) is known as Dirichlet’s kernel.Let𝒞
2
π denote the Banach space of all 2
π-periodic continuous functions defined on [
-
π
,
π
] under the supremum norm. The space L
p
[
0,2
π
] with p
=
∞ reduces to 𝒞
2
π defined over [
0,2
π
]. We write
(4)
∥
f
∥
c
=
sup
⁡
0
≤
t
≤
2
π
|
f
(
t
)
|
,
(
p
=
∞
)
∥
f
∥
p
=
(
∫
0
2
π
‍
|
f
(
t
)
|
p
d
t
)
1
/
p
,
p
≥
1
ω
(
δ
,
f
)
=
sup
⁡
0
≤
|
h
|
≤
δ
∥
f
(
·
+
h
)
-
f
(
·
)
∥
c
,
f
∈
𝒞
2
π
ω
p
(
δ
,
f
)
=
sup
⁡
0
≤
|
h
|
≤
δ
∥
f
(
·
+
h
)
-
f
(
·
)
∥
p
,
ssssssssssssss
f
∈
L
p
[
0,2
π
]
,
p
≥
1
,
when the norm has been taken with respect to x throughout the paper. The quantities ω
(
δ
,
f
) and ω
p
(
δ
,
f
) are, respectively, called the modulus of continuity and integral modulus of continuity of f. It is known ([1], page 45) that ω
(
δ
,
f
) and ω
p
(
δ
,
f
) both tend to zero as δ
→
0.It was Prössdorf [2] who first studied the degree of approximation problems of the Fourier series in H
α
(
0
<
α
≤
1
) space in the Hölder metric. Generalizing the Hölder metric, Leindler [3] introduced the space H
ω given by
(5)
H
ω
=
{
f
∈
𝒞
2
π
:
ω
(
δ
,
f
)
=
O
(
ω
(
δ
)
)
}
,
where ω is a modulus of continuity; that is, ω is a positive nondecreasing continuous function on [
0,2
π
] with the following property: (i)
ω
(
0
)
=
0,
(ii)
ω
(
δ
1
+
δ
2
)
≤
ω
(
δ
1
)
+
ω
(
δ
2
),
(iii)
ω
(
λ
δ
)
≤
(
λ
+
1
)
ω
(
δ
)
,
λ
≥
0.Further Leindler [3] has introduced the following metric on H
ω space:
(6)
∥
f
∥
ω
=
∥
f
∥
c
+
sup
⁡
x
,
y
|
f
(
x
)
-
f
(
y
)
|
ω
(
|
x
-
y
|
)
,
x
≠
y
.In the caseω
(
δ
)
=
δ
α, 0
<
α
≤
1 the space H
ω reduces to H
α space (the norm ∥
·
∥
ω being replaced by ∥
·
∥
α) which is introduced by Prössdorf [2]. It is known that [2]
(7)
H
α
⊆
H
β
⊆
𝒞
2
π
,
0
≤
β
<
α
≤
1
.The degree of approximation problem inH
ω space has been studied by Leindler [3], Totik [4, 5], Mazhar and Totik [6], and Mazhar [7, 8]. The space H
ω was further generalized by Das et al. [9] as follows.Forf
∈
L
p
[
0,2
π
], p
≥
1, we write
(8)
A
(
f
,
ω
)
=
sup
⁡
t
≠
0
∥
f
(
·
+
t
)
-
f
(
·
)
∥
p
ω
(
|
t
|
)
,
where ω is a modulus of continuity. If
(9)
∥
f
(
·
+
t
)
-
f
(
·
)
∥
p
=
O
(
ω
(
|
t
|
)
)
,
then we say that f
∈ Lip(
ω
,
p
). We define
(10)
H
p
(
ω
)
=
{
f
∈
L
p
[
0,2
π
]
,
p
≥
1
:
A
(
f
,
ω
)
<
∞
}
,
(11)
∥
f
∥
p
(
ω
)
=
∥
f
∥
p
+
A
(
f
,
ω
)
.
It can be seen that ∥
f
∥
p
(
ω
) is a norm in H
p
(
ω
). To prove the completeness of the space H
p
(
ω
) we use the completeness of L
p
(
p
≥
1
).If we putω
(
t
)
=
t
α
,
0
<
α
≤
1, then H
p
(
ω
) reduces to H
(
α
,
p
) space (with the norm ∥
f
∥
p
(
ω
) replaced by ∥
f
∥
(
α
,
p
)) which is introduced earlier by Das et al. [10]. If ω
(
t
)
/
t
→
0 as t
→
0, then f
′
(
x
) exists and is 0 everywhere and f is constant. Given the spaces H
p
(
ω
) and H
p
(
υ
), if ω
(
t
)
/
υ
(
t
) is nondecreasing, then
(12)
H
p
(
ω
)
⊆
H
p
(
υ
)
⊆
L
p
,
p
≥
1
,
since
(13)
∥
f
∥
p
(
υ
)
≤
max
⁡
(
1
,
ω
(
2
π
)
υ
(
2
π
)
)
∥
f
∥
p
(
ω
)
.
For 0
≤
β
<
α
≤
1, if we put ω
(
t
)
=
t
α and υ
(
t
)
=
t
β, then (12) reduces to the following:
(14)
H
(
α
,
p
)
⊆
H
(
β
,
p
)
⊆
L
p
,
p
≥
1
.
Note that the space H
(
α
,
∞
) is the familiar H
α space introduced earlier by Prössdorf [2].


### 1.1.  The Cesàro Transformation

Let∑
n
=
0
∞
‍
u
n be an infinite series and let (
s
n
) denote the sequence of its nth partial sums. Then the series ∑
‍
u
n is said to be summable (
C
,
μ
)
(
μ
>
-
1
) to the sum s (finite), if (see [1], page 76)
(15)
σ
n
μ
=
S
n
μ
A
n
μ
⟶
s
as
n
⟶
∞
,
where A
n
μ and S
n
μ are defined by the following formulae:
(16)
∑
n
=
0
∞
‍
A
n
μ
x
n
=
(
1
-
x
)
-
1
-
μ
∑
n
=
0
∞
‍
S
n
μ
x
n
=
(
1
-
x
)
-
μ
∑
n
=
0
∞
‍
s
n
x
n
=
(
1
-
x
)
-
1
-
μ
∑
n
=
0
∞
‍
u
n
x
n
,
where (
|
x
|
<
1
), μ
>
-
1.From the definition ofS
n
μ and A
n
μ it follows that ([1], page 77)
(17)
S
n
μ
=
∑
k
=
0
n
‍
A
n
-
k
μ
-
1
s
k
=
∑
k
=
0
n
‍
A
n
-
k
μ
u
k
A
n
μ
=
(
n
+
μ
μ
)
≃
n
μ
Γ
(
μ
+
1
)
(
μ
≠
-
1
,
-
2
,
…
)
.
The numbers S
n
μ and σ
n
μ are called, respectively, the Cesàro sums and the Cesàro means of order μ
(
μ
>
-
1
) of the series ∑
‍
u
n. Applications of the Cesàro transformation can be found in engineering, for example, modal dynamics in earthquake engineering (see Chen and Hong [11], Chen et al. [12]).


### 1.2. Delayed Arithmetic Mean

Let∑
n
=
0
∞
‍
u
n be an infinite series with sequence of arithmetic mean {
σ
n
}. The Delayed Arithmetic Mean σ
n
,
k of ∑
‍
u
n is given by ([1], page 80)
(18)
σ
n
,
k
=
(
1
+
n
k
)
σ
n
+
k
-
1
-
n
k
σ
n
-
1
,
where k is a positive integer.And it is known [1] that
(19)
(
C
,
1
)
⊂
σ
n
,
k
if
n
k
=
O
(
1
)
as
k
⟶
∞
with
n
.
For k
=
n,
(20)
σ
n
,
n
=
2
σ
2
n
-
1
-
σ
n
-
1
,
which may be calledfirst type Delayed Arithmetic Mean.But in this present paper we takek
=
2
n for Delayed Arithmetic Mean, which is of the form
(21)
σ
n
,
2
n
=
1
2
[
3
σ
3
n
-
1
-
σ
n
-
1
]
.
This may be calledsecond type Delayed Arithmetic Mean.Letσ
n
(
f
;
x
) and σ
n
,
2
n
(
f
;
x
), respectively, denote the first arithmetic mean andsecond type Delayed Arithmetic Mean of (1). It is known that (see [1], page 88 and page 89) the Fejer’s kernel
(22)
F
n
(
t
)
=
2
n
+
1
{
sin
(
1
2
)
(
n
+
1
)
t
2
sin
(
t
2
)
}
2
,
(23)
σ
n
(
f
;
x
)
=
1
π
∫
0
π
‍
F
n
(
t
)
{
f
(
x
+
t
)
+
f
(
x
-
t
)
}
d
t
.
It can be easily verified that
(24)
σ
n
,
2
n
(
f
;
x
)
=
1
n
π
∫
0
π
‍
sin
2
n
t
sin
n
t
(
2
sin
(
t
/
2
)
)
2
{
f
(
x
+
t
)
+
f
(
x
-
t
)
}
d
t
.
We adopt the following additional notations:
(25)
G
x
,
y
(
t
)
=
ϕ
x
+
y
(
t
)
-
ϕ
x
(
t
)
,
l
n
(
x
)
=
σ
n
,
2
n
(
f
;
x
)
-
f
(
x
)
,
h
=
h
(
n
)
=
π
n
.


## 1.1.  The Cesàro Transformation

Let∑
n
=
0
∞
‍
u
n be an infinite series and let (
s
n
) denote the sequence of its nth partial sums. Then the series ∑
‍
u
n is said to be summable (
C
,
μ
)
(
μ
>
-
1
) to the sum s (finite), if (see [1], page 76)
(15)
σ
n
μ
=
S
n
μ
A
n
μ
⟶
s
as
n
⟶
∞
,
where A
n
μ and S
n
μ are defined by the following formulae:
(16)
∑
n
=
0
∞
‍
A
n
μ
x
n
=
(
1
-
x
)
-
1
-
μ
∑
n
=
0
∞
‍
S
n
μ
x
n
=
(
1
-
x
)
-
μ
∑
n
=
0
∞
‍
s
n
x
n
=
(
1
-
x
)
-
1
-
μ
∑
n
=
0
∞
‍
u
n
x
n
,
where (
|
x
|
<
1
), μ
>
-
1.From the definition ofS
n
μ and A
n
μ it follows that ([1], page 77)
(17)
S
n
μ
=
∑
k
=
0
n
‍
A
n
-
k
μ
-
1
s
k
=
∑
k
=
0
n
‍
A
n
-
k
μ
u
k
A
n
μ
=
(
n
+
μ
μ
)
≃
n
μ
Γ
(
μ
+
1
)
(
μ
≠
-
1
,
-
2
,
…
)
.
The numbers S
n
μ and σ
n
μ are called, respectively, the Cesàro sums and the Cesàro means of order μ
(
μ
>
-
1
) of the series ∑
‍
u
n. Applications of the Cesàro transformation can be found in engineering, for example, modal dynamics in earthquake engineering (see Chen and Hong [11], Chen et al. [12]).


## 1.2. Delayed Arithmetic Mean

Let∑
n
=
0
∞
‍
u
n be an infinite series with sequence of arithmetic mean {
σ
n
}. The Delayed Arithmetic Mean σ
n
,
k of ∑
‍
u
n is given by ([1], page 80)
(18)
σ
n
,
k
=
(
1
+
n
k
)
σ
n
+
k
-
1
-
n
k
σ
n
-
1
,
where k is a positive integer.And it is known [1] that
(19)
(
C
,
1
)
⊂
σ
n
,
k
if
n
k
=
O
(
1
)
as
k
⟶
∞
with
n
.
For k
=
n,
(20)
σ
n
,
n
=
2
σ
2
n
-
1
-
σ
n
-
1
,
which may be calledfirst type Delayed Arithmetic Mean.But in this present paper we takek
=
2
n for Delayed Arithmetic Mean, which is of the form
(21)
σ
n
,
2
n
=
1
2
[
3
σ
3
n
-
1
-
σ
n
-
1
]
.
This may be calledsecond type Delayed Arithmetic Mean.Letσ
n
(
f
;
x
) and σ
n
,
2
n
(
f
;
x
), respectively, denote the first arithmetic mean andsecond type Delayed Arithmetic Mean of (1). It is known that (see [1], page 88 and page 89) the Fejer’s kernel
(22)
F
n
(
t
)
=
2
n
+
1
{
sin
(
1
2
)
(
n
+
1
)
t
2
sin
(
t
2
)
}
2
,
(23)
σ
n
(
f
;
x
)
=
1
π
∫
0
π
‍
F
n
(
t
)
{
f
(
x
+
t
)
+
f
(
x
-
t
)
}
d
t
.
It can be easily verified that
(24)
σ
n
,
2
n
(
f
;
x
)
=
1
n
π
∫
0
π
‍
sin
2
n
t
sin
n
t
(
2
sin
(
t
/
2
)
)
2
{
f
(
x
+
t
)
+
f
(
x
-
t
)
}
d
t
.
We adopt the following additional notations:
(25)
G
x
,
y
(
t
)
=
ϕ
x
+
y
(
t
)
-
ϕ
x
(
t
)
,
l
n
(
x
)
=
σ
n
,
2
n
(
f
;
x
)
-
f
(
x
)
,
h
=
h
(
n
)
=
π
n
.


## 2. Introduction

Extensive investigations of approximation by polynomialsS
n and σ
n involve mediocre approximation involving O
(
log
⁡
n
) times worse than the best approximation (see the remarks by [1], page 122). The following theorems, in particular, are being quoted to vindicate that O
(
log
⁡
n
) persists in the order of approximation involving σ
n and S
n.Das et al. [9] developed a new space H
p
(
ω
) and generalized the theorem of Leindler [3] to give the following.Theorem A (see [9]).
Letυ and ω be moduli of continuity such that ω
/
υ is nondecreasing. If f
∈
H
p
(
ω
), p
≥
1, then
(26)
∥
S
n
(
f
;
x
)
-
f
(
x
)
∥
p
(
υ
)
=
O
(
ω
(
π
/
n
)
υ
(
π
/
n
)
log
⁡
n
)
+
O
(
1
)
1
n
∫
π
/
n
π
‍
ω
(
t
)
t
2
υ
(
t
)
d
t
.Theorem B (see [9]).
Iff
∈
Lip
(
ω
,
p
)
,
p
≥
1, and is of monotonic type, then
(27)
∥
S
n
(
f
;
x
)
-
f
(
x
)
∥
p
=
O
(
1
)
1
n
∫
1
/
n
π
‍
ω
(
t
)
t
2
d
t
,
where ω is a modulus of continuity.Recently Leindler [13] proved the following.Theorem C ([13, Theorem 1]).
Letυ and ω be moduli of continuity such that ω
/
υ is nondecreasing; moreover, for some 0
<
ϵ
≤
1, let the function
(28)
γ
(
t
)
:
=
γ
(
υ
,
ω
,
ϵ
;
t
)
:
=
t
-
ϵ
ω
(
t
)
υ
(
t
)
be nonincreasing. If f
∈
H
p
(
ω
), p
≥
1, then
(29)
∥
S
n
-
f
∥
p
(
υ
)
≤
ω
(
1
/
n
)
υ
(
1
/
n
)
log
⁡
n
∀
n
.Leindler [13] has derived Theorem C from Theorem A by making use of the fact that
(30)
∫
π
/
n
π
‍
ω
(
t
)
t
2
υ
(
t
)
d
t
=
O
(
ω
(
π
/
n
)
υ
(
π
/
n
)
log
⁡
n
)
whenever ω
(
t
)
/
t
ϵ
υ
(
t
)
(
0
<
ϵ
≤
1
) is nonincreasing.Quade [14] however proved the following.Theorem D ([14, Theorem 5]).
Letf
∈
Lip
(
α
,
p
)
,
0
<
α
≤
1; then (i)
∥
f
-
S
n
(
f
,
·
)
∥
p
=
O
(
1
/
n
α
)
,
p
>
1,
(ii)
∥
f
-
S
n
(
f
,
·
)
∥
1
=
O
(
(
log
⁡
n
)
/
n
α
)
,
p
=
1.Theorem E ([14, Theorem 6]).
Iff
∈
Lip
(
α
,
p
)
,
0
<
α
≤
1, then(i)
ifp
>
1 or if p
=
1, α
<
1,
(31)
∥
f
-
σ
n
∥
p
=
O
(
1
n
α
)
;
(ii)
ifp
=
α
=
1,
(32)
∥
f
-
σ
n
∥
1
=
O
(
log
⁡
n
n
)
.Prössdorf [2] obtained the following result concerning the degree of approximation of the Fourier series using Fejer’s mean in the Hölder metric.Theorem F (see [2]).
Letf
∈
H
α
(
0
<
α
≤
1
) and 0
≤
β
<
α
≤
1; then
(33)
∥
σ
n
(
f
)
-
f
∥
β
=
O
(
1
)
{
1
n
α
-
β
,
0
<
α
<
1
(
1
+
log
⁡
n
)
1
-
β
n
1
-
β
,
α
=
1
.The objective of the present paper is to establish theorems, which involve Jackson’s order (see [15], page 56) not involving O
(
log
⁡
n
).


## 3. Main Results

We prove the following theorems.Theorem 1.
Letυ and ω be moduli of continuity such that ω
/
υ is nondecreasing. If f
∈
H
p
(
ω
)
,
p
≥
1, then
(34)
(
I
)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
(
υ
)
=
O
(
1
n
)
+
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
υ
(
t
)
d
t
.
If in addition ω
(
t
)
/
t
υ
(
t
) is nonincreasing, then
(35)
(
I
I
)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
(
υ
)
=
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
υ
(
t
)
d
t
and a fortiori
(36)
(
I
I
I
)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
(
υ
)
=
O
(
ω
(
π
/
n
)
υ
(
π
/
n
)
)
.Theorem 2.
Iff
∈
Lip
(
ω
,
p
)
,
p
≥
1, then
(37)
(
I
)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
=
O
(
1
n
)
+
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
d
t
.
If in addition ω
(
t
)
/
t is nonincreasing, then
(38)
(
I
I
)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
=
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
d
t
and a fortiori
(39)
(
I
I
I
)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
=
O
(
ω
(
π
n
)
)
.We need the following lemma.Lemma 3 ([9, Lemma 1]).
Letυ
(
t
) and ω
(
t
) be defined as in Theorem 1. Then for 0
<
t
≤
π and p
≥
1
(i)
∥
ϕ
·
(
t
)
∥
p
≤
2
ω
(
t
),
(ii)
∥
G
·
,
y
(
t
)
∥
p
=
O
(
1
)
{


ω
(
t
)


ω
(
|
y
|
)
,
(iii)
∥
G
·
,
y
(
t
)
∥
p
=
O
(
1
)
υ
(
|
y
|
)
(
ω
(
t
)
/
υ
(
t
)
),
(iv)
∥
G
·
,
y
(
t
)
-
G
·
,
y
(
t
+
h
)
∥
p
=
O
(
1
)
υ
(
|
y
|
)
(
ω
(
h
)
/
υ
(
h
)
), where h
=
π
/
n.


## 4. Proof of Theorem1

It may be easily verified that(40)
2
n
π
∫
0
π
‍
sin
2
n
t
sin
n
t
(
2
sin
(
t
/
2
)
)
2
d
t
=
1
,
and hence
(41)
l
n
(
x
)
=
σ
n
,
2
n
(
f
;
x
)
-
f
(
x
)
=
2
n
π
∫
0
π
‍
ϕ
x
(
t
)
sin
2
n
t
sin
n
t
(
2
sin
(
t
/
2
)
)
2
d
t
.
By definition
(42)
∥
l
n
∥
p
(
ω
)
=
∥
l
n
∥
p
+
sup
⁡
y
≠
0
∥
l
n
(
·
)
-
l
n
(
·
+
y
)
∥
p
ω
(
|
y
|
)
.
Hence
(43)
l
n
(
x
+
y
)
-
l
n
(
x
)
=
2
n
π
∫
0
π
‍
{
ϕ
x
+
y
(
t
)
-
ϕ
x
(
t
)
}
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
d
t
=
2
n
π
∫
0
π
‍
{
G
x
,
y
(
t
)
}
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
d
t
=
2
n
π
∫
0
h
‍
{
G
x
,
y
(
t
)
}
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
d
t
+
2
n
π
∫
h
π
‍
{
G
x
,
y
(
t
)
}
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
d
t
=
I
+
J
.
Since
(44)
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
=
O
(
n
2
)
,
0
<
t
<
π
,
now, by the generalized Minkowski inequality and for p
≥
1, we have
(45)
∥
I
∥
p
≤
2
n
π
∫
0
h
‍
∥
G
·
,
y
(
t
)
∥
p
|
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
|
d
t
=
O
(
1
)
n
υ
(
|
y
|
)
∫
0
h
‍
ω
(
t
)
υ
(
t
)
d
t
,
(
by
Lemma
3
)
=
O
(
1
)
n
υ
(
|
y
|
)
h
ω
(
h
)
υ
(
h
)
=
O
(
1
)
υ
(
|
y
|
)
ω
(
h
)
υ
(
h
)
.
Now,
(46)
J
=
2
n
π
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
(
2
sin
⁡
(
t
/
2
)
)
2
d
t
=
2
n
π
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
{
1
(
2
sin
⁡
(
t
/
2
)
)
2
-
1
t
2
}
d
t
+
2
n
π
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
t
2
d
t
=
J
1
+
J
2
.
Since {
1
/
(
2
sin
⁡
(
t
/
2
)
)
2
-
1
/
t
2
} is bounded and by Lemma 3, we get
(47)
∥
J
1
∥
p
≤
2
n
π
∫
h
π
‍
∥
G
·
,
y
(
t
)
∥
p
d
t
=
O
(
1
)
1
n
υ
(
|
y
|
)
∫
h
π
‍
ω
(
t
)
υ
(
t
)
d
t
=
O
(
1
)
1
n
υ
(
|
y
|
)
.
Now,
(48)
J
2
=
2
n
π
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
t
2
d
t
.
Replacing t with t
+
h in (48), we obtain
(49)
J
2
=
-
2
n
π
∫
0
π
-
h
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
.
From (48) and (49), it follows that
(50)
J
2
=
1
n
π
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
t
2
d
t
-
1
n
π
∫
0
π
-
h
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
=
1
n
π
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
t
2
d
t
-
1
n
π
∫
0
h
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
-
1
n
π
∫
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
+
1
n
π
∫
π
-
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
=
1
n
π
∫
h
π
‍
{
G
x
,
y
(
t
)
t
2
-
G
x
,
y
(
t
+
h
)
(
t
+
h
)
2
}
sin
2
n
t
sin
n
t
d
t
-
1
n
π
∫
0
h
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
+
1
n
π
∫
π
-
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
=
K
-
L
+
M
.
Since sin
2
n
t
sin
n
t
=
O
(
n
2
t
2
), 0
<
t
<
π, it follows that
(51)
∥
L
∥
p
≤
1
n
π
∫
0
h
‍
∥
G
·
,
y
(
t
+
h
)
∥
p
n
2
t
2
(
t
+
h
)
2
d
t
=
O
(
1
)
1
n
υ
(
|
y
|
)
∫
0
h
‍
ω
(
t
+
h
)
υ
(
t
+
h
)
n
2
d
t
=
O
(
1
)
n
υ
(
|
y
|
)
∫
0
h
‍
ω
(
t
+
h
)
υ
(
t
+
h
)
d
t
=
O
(
1
)
n
υ
(
|
y
|
)
ω
(
2
h
)
υ
(
2
h
)
∫
0
h
‍
d
t
=
O
(
1
)
n
υ
(
|
y
|
)
ω
(
h
)
υ
(
h
)
h
=
O
(
1
)
υ
(
|
y
|
)
ω
(
h
)
υ
(
h
)
as ω
(
2
h
)
≤
3
ω
(
h
) and υ
(
2
h
)
≥
υ
(
h
).Since|

sin
2
n
t
sin
n
t
|
≤
1, for t
∈
(
π
-
h
,
π
), it follows that
(52)
∥
M
∥
p
≤
1
n
π
∫
π
-
h
π
‍
∥
G
·
,
y
(
t
+
h
)
∥
p
d
t
(
t
+
h
)
2
=
O
(
1
)
1
n
υ
(
|
y
|
)
∫
π
-
h
π
‍
ω
(
t
+
h
)
υ
(
t
+
h
)
d
t
(
t
+
h
)
2
=
O
(
1
)
1
n
υ
(
|
y
|
)
∫
π
π
+
h
‍
ω
(
θ
)
υ
(
θ
)
d
θ
θ
2
=
O
(
1
)
1
n
2
υ
(
|
y
|
)
.
Now,
(53)
K
=
1
n
π
{
∫
h
π
‍
G
x
,
y
(
t
)
sin
2
n
t
sin
n
t
t
2
d
t
-
∫
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
t
2
d
t
+
∫
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
t
2
d
t
-
∫
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
(
t
+
h
)
2
d
t
}
=
1
n
π
∫
h
π
‍
{
G
x
,
y
(
t
)
-
G
x
,
y
(
t
+
h
)
}
sin
2
n
t
sin
n
t
t
2
d
t
+
1
n
π
∫
h
π
‍
G
x
,
y
(
t
+
h
)
sin
2
n
t
sin
n
t
{
1
t
2
-
1
(
t
+
h
)
2
}
d
t
=
K
1
+
K
2
.
Since |

sin
2
n
t
sin
n
t
|
≤
1, for t
∈
(
h
,
π
), using Lemma 3 (iv), we get
(54)
∥
K
1
∥
p
≤
1
n
π
∫
h
π
‍
∥
G
·
,
y
(
t
)
-
G
·
,
y
(
t
+
h
)
∥
p
d
t
t
2
=
O
(
1
)
1
n
υ
(
|
y
|
)
ω
(
h
)
υ
(
h
)
∫
h
π
‍
d
t
t
2
=
O
(
1
)
υ
(
|
y
|
)
ω
(
h
)
υ
(
h
)
.
Similarly,
(55)
∥
K
2
∥
p
≤
1
n
π
∫
h
π
‍
∥
G
·
,
y
(
t
+
h
)
∥
p
{
1
t
2
-
1
(
t
+
h
)
2
}
d
t
=
O
(
1
)
1
n
υ
(
|
y
|
)
∫
h
π
‍
ω
(
t
+
h
)
υ
(
t
+
h
)
(
t
+
h
)
2
-
t
2
t
2
(
t
+
h
)
2
d
t
=
O
(
1
)
1
n
2
υ
(
|
y
|
)
∫
h
π
‍
ω
(
t
+
h
)
υ
(
t
+
h
)
d
t
t
3
=
O
(
1
)
1
n
2
υ
(
|
y
|
)
∫
h
π
‍
ω
(
t
)
υ
(
t
)
d
t
t
3
.
Collecting the results from (45)–(55), we obtain
(56)
sup
⁡
y
≠
0
∥
l
n
(
·
)
-
l
n
(
·
+
y
)
∥
p
υ
(
|
y
|
)
=
O
(
1
n
)
+
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
υ
(
t
)
d
t
.
Proceeding in the similar lines, it can be proved that
(57)
∥
l
n
∥
p
=
O
(
1
n
)
+
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
d
t
.
Hence collecting results of (42), (56), and (57), we obtain
(58)
∥
l
n
∥
p
(
υ
)
=
O
(
1
n
)
+
O
(
1
)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
υ
(
t
)
d
t
,
which ensures (I) of Theorem 1.If in additionω
(
t
)
/
t
υ
(
t
) is nonincreasing, we have
(59)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
υ
(
t
)
d
t
≥
1
n
2
ω
(
π
)
π
υ
(
π
)
∫
π
/
n
π
‍
d
t
t
2
≥
C
n
,
where C is some positive constant. Further
(60)
1
n
2
∫
π
/
n
π
‍
ω
(
t
)
t
3
υ
(
t
)
d
t
≤
1
n
2
n
π
ω
(
π
/
n
)
υ
(
π
/
n
)
∫
π
/
n
π
‍
d
t
t
2
=
O
(
ω
(
π
/
n
)
υ
(
π
/
n
)
)
.
Using (59) and (60) in (58), we get (II) and (III) of Theorem 1. This completes the proof.


## 5. Proof of Theorem2

We adopt the technique used in the proof of Theorem1 and use the fact that ∥
ϕ
.
(
t
)
∥
p
=
O
(
ω
(
t
)
) whenever f
∈ Lip(
ω
,
p
),p
≥
1, to prove Theorem 2.


## 6. Remarks and Corollaries

Comparing Theorem1 with Theorem A and Theorem C we observe that trigonometric polynomial σ
n
,
2
n
(
f
;
x
) provides sharper estimation than those obtained by sequence of partial sums of a Fourier series. The comparison of Theorem 2 with Theorem B also exhibits the advantage of σ
n
,
2
n
(
f
;
x
) over the sequence of partial sums of the Fourier series in the degree of approximation problem.We specializeω
(
t
) and υ
(
t
) to obtain the following corollaries from Theorems 1 and 2. Taking ω
(
t
)
=
t
α and υ
(
t
)
=
t
β, 0
≤
β
<
α
≤
1 in Theorem 1, we get the following.Corollary 4.
Iff
∈
H
(
α
,
p
)
,
p
≥
1,0
≤
β
<
α
≤
1, then
(61)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
(
β
,
p
)
=
O
(
1
n
α
-
β
)
.Takingp
=
∞ in Corollary 4, we obtain the following.Corollary 5.
Iff
∈
H
α, 0
≤
β
<
α
≤
1, then
(62)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
β
=
O
(
1
n
α
-
β
)
.The above corollary can be compared with Theorem F.Takingω
(
t
)
=
t
α, 0
<
α
≤
1 in Theorem 2, we obtain the following.Corollary 6.
Iff
∈
L
i
p
(
α
,
p
), 0
<
α
≤
1 and p
≥
1, then
(63)
∥
σ
n
,
2
n
(
f
;
·
)
-
f
(
·
)
∥
p
=
O
(
1
n
α
)
.For the space Lip(
1,1
) Quade has provided the result (32) of Theorem E using σ
n
(
f
;
x
) trigonometric polynomial. Estimate given in (32) certainly is not of Jackson’s order. Therefore in Corollary 6 a more general trigonometric polynomial (namely, σ
n
,
2
n
(
f
;
x
)) has been taken to ensure Jackson’s order, which is sharper than the one given in (32).



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*Source: 171675-2014-05-07.xml*