Journal of Inequalities in Pure and
Applied Mathematics
SOME RESULTS ON L1 APPROXIMATION OF THE r-TH DERIVATE
OF FOURIER SERIES
volume 3, issue 1, article 10,
2002.
Z. TOMOVSKI
Faculty of Mathematical and Natural Sciences
P.O. Box 162
91000 Skopje
MACEDONIA
EMail: tomovski@iunona.pmf.ukim.edu.mk
Received 22 September, 1999;
received in final form 17 October, 2001;
accepted 18 October, 2001.
Communicated by: A.G. Babenko
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper we obtain the conditions for L1 -convergence of the r-th derivatives of the cosine and sine trigonometric series. These results are extensions
of corresponding Sidon’s and Telyakovskii’s theorems for trigonometric series
(case: r = 0).
2000 Mathematics Subject Classification: 26D15, 42A20.
Key words: L1 -approximation, Fourier series, Sidon-Telyakovskii class, Telyakovskii
inequality
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
The author would like to thank the referee for his suggestions and encouragement.
Živorad Tomovski
Contents
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4
Proofs of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References
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1.
Introduction
Let
∞
a0 X
f (x) =
+
an cos nx ,
2
n=1
(1.1)
g(x) =
(1.2)
∞
X
an sin nx
n=1
be the cosine and sine trigonometric series with real coefficients.
Let ∆an = an − an+1 , n ∈ {0, 1, 2, 3, . . .}. The Dirichlet’s kernel, conjugate
Dirichlet’s kernel and modified Dirichlet’s kernel are denoted respectively by
n
sin n + 12 t
1 X
Dn (t) = +
cos kt =
,
2 k=1
2 sin 2t
n
X
cos 2t − cos n +
D̃n (t) =
sin kt =
2 sin 2t
k=1
1
2
t
,
cos n + 12 t
1
t
Dn (t) = − ctg + D̃n (t) = −
.
2
2
2 sin 2t
Let
En (t) =
n
X
n
X
1
1
+
eikt and E−n (t) = +
e−ikt .
2 k=1
2 k=1
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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(r)
(r)
Then the r-th derivatives Dn (t) and D̃n (t) can be written as
(r)
(1.3)
2Dn(r) (t) = En(r) (t) + E−n (t) ,
(1.4)
2iD̃n(r) (t) = En(r) (t) − E−n (t) .
(r)
In [2], Sidon proved the following theorem.
∞
∞
Theorem 1.1. Let
P∞{αn }n=1 and {pn }n=1 be sequences such that |αn | ≤ 1, for
every n and let n=1 |pn | converge. If
(1.5)
an =
∞
k
X
pk X
k=n
k
αl , n ∈ N
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
l=n
then the cosine series (1.1) is the Fourier series of its sum f.
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Several authors have studied the problem of L1 −convergence of the series
(1.1) and (1.2).
In [4] Telyakovskii defined the following class of L1 -convergence of Fourier
series. A sequence {ak }∞
k=0 belongs to the class S, or {ak } ∈ S if ak → 0 as
k →P∞ and there exists a monotonically decreasing sequence {Ak }∞
k=0 such
that ∞
A
<
∞
and
|∆a
|
≤
A
for
all
k.
k
k
k=0 k
The importance of Telyakovskii’s contributions are twofold. Firstly, he expressed Sidon’s conditions (1.5) in a succinct equivalent form, and secondly, he
showed that the class S is also a class of L1 -convergence. Thus, the class S is
usually called the Sidon–Telyakovskii class.
In the same paper, Telyakovskii proved the following two theorems.
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Theorem 1.2. [4]. Let the coefficients of the series f (x) belong to the class S.
Then the series is a Fourier series and the following inequality holds:
Z π
∞
X
|f (x)| dx ≤ M
An ,
0
n=0
where M is a positive constant, independent on f .
Theorem 1.3. [4]. Let the coefficients of the series g (x) belong to the class S.
Then the following inequality holds for p = 1, 2, 3, . . .
!
Z π
p
∞
X
X
|an |
|g (x)| dx =
+O
An .
n
π/(p+1)
n=1
n=1
In particular, g (x) is a Fourier series iff
P∞
n=1
|an |
n
< ∞.
In [5], we extended the Sidon–Telyakovskii class S = S0 , i.e., we defined
the class Sr , r = 1, 2, 3, . . . as follows: {ak }∞
k=1 ∈ Sr if ak → 0 as k →
∞
and
there
exists
a
monotonically
decreasing
sequence {Ak }∞
k=1 such that
P∞ r
Ak for all k.
k=1 k Ak < ∞ and |∆ak | ≤ P
r
We note that by Ak ↓ 0 and ∞
k=1 k Ak < ∞, we get
(1.6)
k r+1 Ak = o (1) ,
k → ∞.
It is trivially to see that Sr+1 ⊂ Sr for all r = 1, 2, 3, .... Now, let {an }∞
n=1 ∈
∞
S1 . For arbitrary real number a0 , we shall prove that sequence {an }n=0 belongs
to S0 . We define A0 = max(|∆a0 |, A1 ). Then |∆a0 | ≤ A0 , i.e. |∆an | ≤ An ,
for all n ∈ {0, 1, 2, ...} and {An }∞
n=0 is monotonically decreasing sequence.
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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On the other hand,
∞
X
An ≤ A0 +
n=0
∞
X
nAn < ∞ .
n=1
Thus, {an }∞
n=0 ∈ S0 , i.e. Sr+1 ⊂ Sr , for all r = 0, 1, 2, . . .. The next example
verifies that the implication
{an } ∈ Sr+1 ⇒ {an } ∈ Sr ,
r = 0, 1, 2, . . .
is not reversible.
P
1
Example 1.1. For n = 0, 1, 2, 3, ... define an = ∞
k=n+1 k2 . Then an → 0 as
1
n → ∞ and for n = 0, 1, 2, 3, . . ., ∆an = (n+1)2 . Firstly we shall show that
{an }∞
/ S1 .
n=1 ∈
Let {An }∞
positive
such that A ↓ 0 and ∆an =
n=1 is an arbitrary
P∞
P∞sequence
n
|∆an | ≤ An . However, n=1 nAn ≥ n=1 (n+1)2 is divergent, i.e. {an } ∈
/ S1 .
1
Now, for all n = 0, 1, 2, . . . let An = (n+1)2 . Then An ↓ 0, |∆an | ≤ An and
P∞
P∞ 1
∞
n=0 An =
n=1 n2 < ∞, i.e. {an }n=0 ∈ S0 .
Our next example will show that there exists a sequence {an }∞
n=1 such that
∞
∞
{an }n=1 ∈ Sr but {an }n=1 ∈
/ Sr+1 , for all r = 1, 2, 3, . . ..
P
1
Namely, for all n = 1, 2, 3, . . . let an = ∞
k=n kr+2 . Then an → 0 as n → ∞
1
and for n = 1, 2, 3, . . . , ∆an = nr+2
. Let {An }∞
n=1 is an arbitrary positive
sequence such that An ↓ 0 and ∆an = |∆an | ≤ An . However,
∞
X
n=1
nr+1 An ≥
∞
X
n=1
nr+1
1
nr+2
∞
X
1
=
n
n=1
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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is divergent, i.e. {an } ∈
/ Sr+1 . On the other
for all
= 1, 2, . . . let
P∞hand,
Pn
∞
1
r
An = nr+2 . Then An ↓ 0, |∆an | ≤ An and n=1 n An = n=1 n12 < ∞, i.e.
{an } ∈ Sr .
In the same paper [5] we proved the following theorem.
Theorem 1.4. [5]. Let the coefficients of the series (1.1) belong to the class
Sr , r = 0, 1, 2, .... Then the r−th derivative of the series (1.1) is a Fourier
series of some f (r) ∈ L1 (0, π) and the following inequality holds:
Z
π
∞
X
(r)
f (x) dx ≤ M
n r An ,
0
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
n=1
Živorad Tomovski
where 0 < M = M (r) < ∞.
This is an extension of the Telyakovskii Theorem 1.2.
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2.
Results
In this paper, we shall prove the following main results.
Theorem 2.1. A null sequence {an } belongs to the class Sr , r = 0, 1, 2, . . . if
and only if it can be represented as
an =
(2.1)
∞
k
X
pk X
k=n
k
αl , n ∈ N
l=n
∞
where {αn }∞
n=1 and {pn }n=1 are sequences such that |αn | ≤ 1, for all n and
∞
X
nr |pn | < ∞.
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
n=1
Corollary 2.2.
every n and let
∞
Let {αn }∞
n=1 and {pn }n=1 be sequences
P
∞
r
n=1 n |pn | < ∞, r = 0, 1, 2, . . . . If
an =
∞
k
X
pk X
k=n
k
such that |αn | ≤ 1, for
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l=n
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then the r−th derivate of the series (1.1) is a Fourier series of some f (r) ∈ L1 .
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Theorem 2.3. Let the coefficients of the series g (x) belong to the class Sr , r =
0, 1, 2, . . . Then the r-th derivate of the series (1.2) converges to a function and
for m = 1, 2, 3, . . . the following inequality holds:
!
Z π
m
∞
X
X
(r)
r−1
r
g (x) dx ≤ M
(∗)
|an | · n
+
n An ,
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n=1
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where
0 < M = M (r) < ∞ .
P∞
Moreover, if n=1 nr−1 |an | < ∞, then the r-th derivate of the series (1.2) is a
Fourier series of some g (r) ∈ L1 (0, π) and
!
Z π
∞
∞
X
X
(r)
r−1
r
g (x) dx ≤ M
|an | · n
+
n An
0
n=1
n=1
Corollary 2.4. Let the coefficients of the series g (x) belong to the class Sr ,
r ≥ 1. Then the following inequality holds:
Z
π
∞
X
(r)
g (x) dx ≤ M
n r An ,
0
where 0 < M = M (r) < ∞.
n=1
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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3.
Lemmas
For the proof of our new theorems we need the following lemmas.
The following lemma proved by Sheng, can be reformulated in the following
way.
Lemma 3.1. [1] Let r be a nonnegative integer and x ∈ (0, π], where n ≥ 1.
Then
r
kπ
1 k
1
X
n
+
sin
n
+
x
+
2
2
2
ϕk (x),
Dn(r) (x) =
r+1−k
x
sin 2
k=0
where ϕr ≡ 12 and ϕk , k = 0, 1, 2, . . . , r − 1 denotes various entire 4π –
periodic functions of x, independent of n. More precisely, ϕk , k = 0, 1, 2, . . . , r
are trigonometric polynomials of x2 .
Lemma 3.2. Let {αj }kj=0 be a sequence of real numbers. Then the following
relation holds for ν = 0, 1, 2, . . . , r and r = 0, 1, 2, . . .
Z π
k
X
1 ν
1
ν+3
j
+
sin
j
+
x
+
π
2
2
2
Uk =
αj
dx
r+1−ν
x
π/(k+1) j=0
sin 2

!1/2 
k
X
1
.
= O (k + 1)r−ν+ 2
αj2 (j + 1)2ν
j=0
Proof. Applying first Cauchy–Buniakowski inequality, yields
"Z
#1/2
π
dx
Uk ≤
2(r+1−ν)
π/(k+1) sin x
2
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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×

Z

"
π
π/(k+1)
k
X
αj j +
j=0
1
2
ν

#2 1/2
1
(ν + 3)π
sin j +
x+
dx
.

2
2
Since
Z
π
π/(k+1)
sin
dx
≤ π 2(r+1−ν)
2(r+1−ν)
x
2
Z
π
π/(k+1)
dx
x2(r+1−ν)
π(k + 1)2(r+1−ν)−1
2(r + 1 − ν) − 1
≤ π(k + 1)2(r+1−ν)−1 ,
≤
we have
1/2
Uk ≤ π(k + 1)2(r+1−ν)−1


" k
ν
#2 1/2
Z π X
1
1
ν+3
×
αj j +
sin j +
x+
π
dx

 0
2
2
2
j=0
1/2
≤ 2π(k + 1)2(r+1−ν)−1


" k
ν
#2 1/2
Z 2π X
ν+3
1
×
sin (2j + 1)t +
π
dt
.
αj j +

 0
2
2
j=0
Then, applying Parseval’s equality, we obtain:
" k
#1/2
h
i1/2 X
Uk ≤ 2π (k + 1)2(r+1−ν)−1
|αj |2 (j + 1)2ν
.
j=0
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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Finally,

1
Uk = O (k + 1)r−ν+ 2
k
X
!1/2 
αj2 (j + 1)2ν
.
j=0
Lemma 3.3. Let r ∈ {0, 1, 2, 3, . . . } and {αk }nk=0 be a sequence of real numbers such that |αk | ≤ 1, for all k. Then there exists a finite constant M =
M (r) > 0 such that for any n ≥ 0
n
Z π
X
(r)
(∗∗)
αk Dk (x) dx ≤ M · (n + 1)r+1 .
π/(n+1)
k=0
Proof. Similar to Lemma 3.1 it is not difficult to proof the following equality
r
X
n + 12 k sin n + 12 x + k+3
π
(r)
2
Dn (x) =
ϕk (x),
r+1−k
sin x2
k=0
where ϕk denotes the same various 4π-periodic functions of x, independent of
n.
Now, we have:
n
Z π
X
(r)
αk Dk (x) dx
π/(n+1) k=0
!
Z π
n
r
X
X
π
j + 12 ν sin j + 12 x + ν+3
2
≤
αj
ϕ
(x)
dx.
ν
r+1−ν
π/(n+1) j=0
sin x2
ν=0
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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Since ϕν are bounded, we have:
ν
Z π
n
X
j+ 12 sin j + 21 x +
αj
r+1−ν
π/(n+1) sin x
j=0
ν+3
π
2
2
ϕν (x) dx ≤ K Un ,
where Un is the integral as in Lemma 3.2, and K = K(r) is a positive constant.
Applying Lemma 3.2, to the last integral, we obtain:
Z π
n
X
1
ν+3
1 ν
j + 2 sin j + 2 x + 2 π
αj
ϕν (x) dx
r+1−ν
x
π/(n+1) j=0
sin 2

!1/2 
n
X
1

= O (n + 1)r−ν+ 2
αj2 (j + 1)2ν
j=0
= O (n + 1)
r−ν+ 12
(n + 1)
ν+ 12
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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= O (n + 1)
r+1
.
Finally the inequality (∗∗) is satisfied.
Remark 3.1. For r = 0, we obtain the Telyakovskii type inequality, proved in
[4].
Lemma 3.4. Let r be a non-negative integer. Then for all 0 < |t| ≤ π and all
n ≥ 1 the following estimates hold:
(r) 4nr π
(i) E−n (t) ≤ |t| ,
r
(r) (ii) D̃n (t) ≤ 4n|t|π ,
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(r) (iii) Dn (t) ≤
4nr π
|t|
+O
1
|t|
r+1
.
Proof. (i) The case r = 0 is trivial. Really,
|En (t)| ≤ |Dn (t)| + |D̃n (t)| ≤
π
π
3π
4π
+
=
<
,
2|t| |t|
2|t|
|t|
|E−n (t)| = |En (−t)| <
4π
.
|t|
Let r ≥ 1. Applying the Abel’s transformation, we have:
" n−1
#
n
X
X
1
1
En(r) (t) = ir
k r eikt = ir
∆(k r ) Ek (t) −
+ nr En (t) −
2
2
k=1
k=1
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
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n−1
X
1
1
(r)
r
r
r
|En (t)| ≤
[(k + 1) − k ]
+ |Ek (t)| + n
+ |En (t)|
2
2
k=1
)
(X
n−1
3π
4πnr
π
+
[(k + 1)r − k r ] + nr =
.
≤
2|t| 2|t|
|t|
k=1
(r) 4nr π
(r)
(r)
Since E−n (t) = En (−t), we obtain E−n (t) ≤ |t| .
(ii) Applying the inequality (i), we obtain
(r) (r) 1 (r) 1 (r) 4nr π
D̃
(t)
=
i
D̃
(t)
≤
E
(t)
+
E
(t)
.
n
n
≤
2 n
2 −n
|t|
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(r) 1
(iii) We note that ctg 2t
. Applying the inequality (ii), we
= O |t|r+1
obtain
(r) 1
t
1
(r)
4nr π
(r)
|Dn (t)| ≤ |D̃n (t)| + ctg
+O
.
≤
2
2
|t|
|t|r+1
Some Results on
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Derivate of Fourier Series
Živorad Tomovski
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4.
Proofs of the Main Results
Proof of Theorem 2.1. Let (2.1) hold. Then
∞
X
pm
∆ak = αk
,
m
m=k
and we denote
∞
X
|pm |
Ak =
.
m
m=k
Some Results on
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Derivate of Fourier Series
Since |αk | ≤ 1, we get
Živorad Tomovski
|∆ak | ≤ |αk |
∞
X
|pm |
≤ Ak , for all k .
m
m=k
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However,
∞
X
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∞
∞
m
∞
X
|pm | X |pm | X r X r
k Ak =
k
=
k ≤
m |pm | < ∞ ,
m
m k=1
m=1
m=1
k=1
k=1
m=k
r
∞
X
r
and Ak ↓ 0 i.e. {ak } ∈ Sr .
k
Now, if {ak } ∈ Sr , we put αk = ∆a
and pk = k (Ak − Ak+1 ) .
Ak
Hence |αk | ≤ 1, and by (1.6) we get:
∞
X
k=1
k r |pk | =
∞
X
k=1
k r+1 (Ak − Ak+1 ) ≤
∞
X
k=1
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(r + 1) k r Ak < ∞ .
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Finally,
ak =
=
∞
X
i=k
∞
X
∆ai =
∞
X
αi Ai =
i=k
∞
X
pm
αi
m=i
i=k
m
=
∞
X
∞
X
αi
∞
X
∆Am
m=i
i=k
m
pm X
αi ,
m=k
m
i=k
i.e. (2.1) holds.
Proof of Corollary 2.2. The proof of this corollary follows from Theorems 1.4
and 2.1.
Proof of Theorem 2.3. We suppose that a0 = 0 and A0 = max (|a1 | , A1 ) .
Applying the Abel’s transformation, we have:
(4.1)
g(x) =
∞
X
∆ak Dk (x) ,
x ∈ (0, π] .
P
(r)
Applying Lemma 3.4 (iii), we have that the series ∞
k=1 ∆ak D k (x) is uniformly convergent on any compact subset of [ε, π], where ε > 0.
Thus, representation (4.1) implies that
g
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k=0
(r)
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
(x) =
∞
X
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(r)
∆ak Dk (x) .
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k=0
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Then,
Zπ
|g (r) (x)|dx ≤
j=1
π/(m+1)
j−1
X
(r)
∆ak Dk (x) dx
π/(j+1) k=0
!
m Z π/j
∞
X
X
(r)
+O
∆ak Dk (x) dx .
π/(j+1) m Z
X
π/j
j=1
k=j
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Let
j−1
X
(r)
I1 =
∆ak Dk (x) dx ,
j=1 π/(j+1) k=0
m Z π/j
∞
X
X
(r)
D
(x)
I2 =
∆a
dx .
k k
π/(j+1)
m Z
X
π/j
j=1
Since ctg x2 =
ing estimate
2
x
k=j
+
P∞
x (r) 2(−1)r r!
ctg
=
+ O(1), x ∈ (0, π] .
2
xr+1
4x
n=1 x2 −4n2 π 2
(see [3]) it is not difficult to proof the follow-
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Thus
(r)
Dn (x) =
r+1
(−1) r!
+ O (n + 1)r+1 , x ∈ (0, π]
r+1
x
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Hence,
j−1
Z
m X
π/j
X
dx
I1 = r!
∆ak π/(j+1) xr+1
j=1 k=0
#Z
" j−1
m
X
X
+O
|∆ak |(k + 1)r+1
j=1
= Or
m
X
|aj |j r−1
+O
j=1
dx
π/(j+1)
k=0
!
!
π/j
j−1
m X
X
(k + 1)r+1 |∆ak |
j(j + 1)
j=1 k=0
!
,
Živorad Tomovski
where Or depends on r. However,
j−1
m X
X
(k + 1)r+1 |∆ak |
j=1 k=0
j(j + 1)
=
≤
m
X
j=1
∞
X
j−1
X
1
(k + 1)r+1 |∆ak |
j(j + 1) k=0
(k + 1)r+1 |∆ak |
k=0
=
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
∞
X
∞
X
1
j(j + 1)
j=k+1
(k + 1)r |∆ak |
k=0
= |∆a0 | +
≤ |a1 | + 2r
∞
X
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(k + 1)r |∆ak |
k=1
∞
X
k=1
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k r |∆ak |
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≤
∞
X
|∆ak | + 2r
∞
X
k r Ak ≤ (1 + 2r )
k=1
k=1
∞
X
k r Ak .
k=1
Thus,
j−1
m X
X
|∆ak |(k + 1)r+1
j(j + 1)
j=1 k=0
= Or
∞
X
!
k r Ak
,
k=1
where Or depends on r.
Therefore,
m
X
I1 = Or
!
|aj |j
r−1
+ Or
j=1
∞
X
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
!
k r Ak
.
Živorad Tomovski
k=1
Application of Abel’s transformation, yields
∞
X
(r)
∆ak Dk (x)
=
∞
X
k=j
k=j
∆Ak
k
X
∆ai
i=0
Ai
(r)
Di (x)
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− Aj
j−1
X
∆ai
i=0
Ai
(r)
Di (x) .
Let us estimate the second integral:
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k
m
∞
X ∆a
π
X
X
i (r)
D
(x)
I2 ≤
(∆Ak )
i
A
i
π/(j+1) i=0
j=1 k=j
"
Z
#
Z π/j X
j−1
∆ai (r) +Aj
Di (x) dx .
π/(j+1) i=0 Ai
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Applying the Lemma 3.3, we have:
k
Z π
X ∆a
(r)
i
(4.2)
Jk =
Di (x) dx = Or (k + 1)r+1 ,
π/(j+1) i=0 Ai
where Or depends on r. Then, by Lemma 3.4(iii),
Z π/j X
j−1
∆a
i (r)
D
(x)
dx|
i
A
i
π/(j+1) i=0
!!
!
Z
Z
j−1
j−1
X
X
|∆ai | π/j dx
|∆ai | π/j
dx
r
=O j
+O
Ai π/(j+1) x
Ai π/(j+1) xr+1
i=0
i=0
(4.3)
= O(j r ) + Or (j r ) = Or (j r )
j=1
= Or (1)
∞
X
(∆Ak )(k + 1)r+1 + Or
= Or
∞
X
!
j r Aj
∞
X
j=1
k=1
Živorad Tomovski
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where Or depends on r. However, by (4.2), (4.3) and (1.6), we have
!
∞
∞
X
X
I2 ≤
(∆Ak ) Jk + Or
j r Aj
k=1
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L1 -Approximation of the r-th
Derivate of Fourier Series
!
j r Aj
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.
j=1
Finally, the inequality (∗) is satisfied.
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Proof of Corollary 2.4. By the inequalities
m
X
|an | · n
r−1
≤
n=1
≤
∞
X
n=1
∞
X
n
r−1
nr−1
n=1
=
∞
X
Ak
k=1
∞
X
|∆ak |
k=n
∞
X
k=n
k
X
n
Ak
r−1
≤
n=1
∞
X
k=1
and Theorem 2.3, we obtain:
Z
π
(r)
|g (x)|dx ≤ M
0
where 0 < M = M (r) < ∞.
k r Ak ,
Some Results on
L1 -Approximation of the r-th
Derivate of Fourier Series
Živorad Tomovski
∞
X
n=1
!
n r An
,
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References
[1] SHENG SHUYUN, The extensions of the theorem of C. V. Stanojevic and
V.B. Stanojevic, Proc. Amer. Math. Soc., 110 (1990), 895–904.
[2] S. SIDON, Hinrehichende Bedingungen fur den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc., 72 (1939), 158–160.
[3] M.R. SPIEGEL, Theory and Problems of Complex Variables, Singapore
(1988), p.175, 192.
[4] S.A. TELYAKOVSKII, On a sufficient condition of Sidon for the integrability of trigonometric series, Math. Notes, 14 (1973), 742–748.
[5] Ž. TOMOVSKI, An extension of the Sidon–Fomin inequality and applications, Math. Ineq. & Appl., 4(2) (2001), 231–238.
[6] Ž. TOMOVSKI, Some results on L1 -approximation of the r-th derivate of
Fourier series, RGMIA Research Report Collection, 2(5) (1999), 709–717.
ONLINE: http://rgmia.vu.edu.au/v2n5.html
Some Results on
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