ON A GENERALIZATION OF LIPSCHITZ’S
CLASSES
Generalization of Lipschitz’s
Classes
Xh. Z. KRASNIQI
Department of Mathematics and Computer Sciences,
Avenue "Mother Teresa" 5
Prishtinë, 10000, Republic of Kosovo
EMail: xheki00@hotmail.com
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
Title Page
Contents
Received:
26 March, 2008
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Accepted:
06 August, 2008
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Communicated by:
H. Bor
2000 AMS Sub. Class.:
42A20, 42A32.
Key words:
Lipschitz classes, Fourier series.
Abstract:
In this paper we obtain a generalization of Lipschitz’s classes Λm (β, p, r) defined in [1]. We give necessary conditions for even or odd functions with Fourier
series to belong to the classes Λm (p, r, α). We also give sufficient conditions for
even or odd functions with Fourier series to belong to the same classes.
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Contents
1
Definitions and Useful Statements
3
2
Main Results
7
Generalization of Lipschitz’s
Classes
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
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Contents
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1.
Definitions and Useful Statements
We consider the series
∞
a0 X
+
an cos nx
2
n=1
(1.1)
or
∞
X
(1.2)
Generalization of Lipschitz’s
Classes
an sin nx
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
n=1
where an are Fourier coefficients of integrable function f .
Definition 1.1. We say that a function f belongs to W Ap , (1 < p < ∞) if
!p
∞
∞
X
X
np−2
|∆ak | < +∞
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where ∆ak = ak − ak+1 (see [1]).
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We say that any function α(t) is a function of type σ (see [4]) if it is measurable in
[0, 1], integrable in [δ, 1] for each δ ∈ (0, 1), and there exist real numbers C1,α > 0,
σ and δ0 ∈ (0, 1) such that
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n=1
k=n
1. α(t) ≥ C1,α , for all t ∈ [0, 1];
Rδ
2. 0 α(t)ts dt < ∞ for each s > σ and δ ∈ (0, δ0 );
Rδ
3. 0 α(t)ts dt = ∞ for each s < σ and δ ∈ (0, δ0 ), and
Z δ
Z 2δ
σ
σ
α(t)t dt ≤ C2 δ
α(t)dt.
0
δ
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In [1], the classes Λm (β, p, r, ) are defined in the following way
Definition 1.2. f ∈ Λm (β, p, r, ), if
(m)
f ≡
β,p,r
(Z
1
2π
Z
0
0
|∆m f (x, t)|p
dx
tβp
pr
dt
t
) r1
< +∞,
Generalization of Lipschitz’s
Classes
where 1 < p < +∞, 1 ≤ r < +∞, β > 0, m ∈ N and
∆m f (x, t) =
m
X
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
i
f [x + (m − 2i)t].
(−1)i Cm
i=1
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Now we define classes Λm (p, r, α) as follows:
Contents
Definition 1.3. We say f ∈ Λm (p, r, α), if
(m)
f ≡
p,r,α
(Z
1
Z
p
|∆m f (x, t)| dx
α(t)
0
2π
pr
) r1
dt
< +∞,
0
where α(t) is function of the type σ.
For α(t) = t−rβ−1 , β > 0, we get the classes Λm (β, p, r), considered in [1].
Therefore classes Λm (p, r, α) are generalizations of classes Λm (β, p, r).
We need some auxiliary statements.
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Lemma
1.4 ([2]). Let aν , bν and βn be numbers such that aν ≥ 0, bν ≥ 0 and
P∞
a
ν=n ν = an βn :
1. For 0 < p ≤ 1 the following inequality is valid
!p
∞
ν
∞
X
X
X
p
aν
bµ
≥p
aν (bν βν )p ;
ν=1
µ=1
Generalization of Lipschitz’s
Classes
ν=1
Xh. Z. Krasniqi
2. For 1 ≤ p < ∞ we have
vol. 9, iss. 3, art. 73, 2008
∞
X
aν
ν=1
ν
X
!p
≤ pp
bµ
∞
X
aν (bν βν )p .
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ν=1
µ=1
Lemma 1.5 ([2]). Let aν , bν and γn be numbers such that aν ≥ 0, bν ≥ 0 and
P
n
ν=1 aν = bn γn :
1. For 0 < p ≤ 1, we have
∞
X
aν
ν=1
∞
X
!p
≥p
bµ
p
µ=ν
∞
X
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p
aν (bν γν ) ;
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ν=1
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2. For 1 ≤ p < ∞, we have
∞
X
ν=1
aν
Close
∞
X
µ=ν
!p
bµ
≤ pp
∞
X
ν=1
aν (bν γν )p .
Lemma 1.6 ([3]). Let µ, τ and aν be numbers such that 0 < µ < τ < ∞ and
aν ≥ 0. Then
! τ1
! µ1
∞
∞
X
X
aτν
≤
aµν
.
ν=1
ν=1
We denote by C a constant that depends only on m, p, r and may be different in
different relations.
Theorem 1.7 ([1]). If f ∈ W Ap , 1 < p < +∞, then
!p
∞
X
X
X
(m)
p
mp
(m+1)p−2
n
|∆ak | +C
np−2
ωp (h; f ) ≤ Ch
1
1
k=n
n≤[ h
n>[ h
]
]
Generalization of Lipschitz’s
Classes
Xh. Z. Krasniqi
∞
X
k=n
vol. 9, iss. 3, art. 73, 2008
!p
|∆ak |
,
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Contents
(m)
where ωp (h; f ) is the integral modulus of smoothness of order m.
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2.
Main Results
Let us denote
1/n
Z
A(n) :=
α(t)dt,
1/(n+1)
b(n) := b1 (n) + b2 (n) = n
mr
1/n
Z
mr
α(t)t
Z
1
dt +
α(t)dt.
0
Generalization of Lipschitz’s
Classes
1/(n+1)
Xh. Z. Krasniqi
We have the following first main result.
vol. 9, iss. 3, art. 73, 2008
Theorem 2.1. Let m be any natural number and
f ∈ AW p ,
1 ≤ r < +∞.
P
If for the coefficients of series (1.1) or (1.2) we have ∞
k=1 |∆ak | < +∞, then:
1 < p < +∞,
1. For p ≤ r we have
(m)
f ≤C
p,r,α
(∞
X
nr
1− p2
∞
X
∆ak n=1
!r
b(n)
k=n
b(n)
A(n)
pr −1 ) r1
;
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2. For p > r we have
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(
(m)
f ≤C
p,r,α
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∞
X
n=1
n
r
1− p2
∞
X
∆ak k=n
!r
) r1
b(n)
Close
.
Proof. Using the characteristics of the integral modulus of smoothness we have
Z 2π
pr
n or Z 1
(m)
p
f =
α(t)
|∆m f (x, t)| dx dt
p,r,α
0
≤
≤
=
0
∞ Z
X
N =1
∞
X
N =1
∞
X
1/N
r
α(t) ωp(m) (f ; t) dt
1/(N +1)
(m)
r
ωp (f ; 1/N )
Z
Generalization of Lipschitz’s
Classes
1/N
α(t)dt
Xh. Z. Krasniqi
1/(N +1)
vol. 9, iss. 3, art. 73, 2008
r
A(N ) ωp(m) (f ; 1/N ) .
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N =1
Contents
According to the Theorem 1.7, we have
∞
n or
X
f (m)
≤
C
A(N )N −mr
p,r,α
( N
X
n
(m+1)p−2
n=1
N =1
+C
∞
X
N =1
!p ) pr
|∆ak |
∞
X
np−2
n=N +1
∞
X
!p ) pr
|∆ak |
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k=n
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= I1 + I2 .
Now we estimate I1 and I2 . Let r/p ≥ 1. Then according to Lemma 1.4 we have
(
!p ) pr
∞
∞
X
X
I1 ≤ C
A(N )N −mr N (m+1)p−2
|∆ak | βN
.
N =1
JJ
k=n
(
A(N )
∞
X
k=N
Close
Now we estimate the quantity βN :
A(N )N
−mr
βN =
=
∞
X
A(i)i−mr
i=N
∞ X
i=N
≤ 2mr
i+1
i
Z
mr
1
·
(i + 1)mr
Z
1/i
α(t)dt
1/(i+1)
Generalization of Lipschitz’s
Classes
1/N
α(t)tmr dt,
Xh. Z. Krasniqi
0
vol. 9, iss. 3, art. 73, 2008
or
βN ≤ C
b1 (N )
.
A(N )
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Contents
Consequently
(2.1)
I1 ≤ C
∞
X
A(N )N
r (1− p2 )
N =1
∞
X
!r |∆ak |
k=N
b1 (N )
A(N )
pr
.
According to Lemma 1.5, for r/p ≥ 1 we have
(
!p ) pr
∞
∞
X
X
I2 ≤ C
A(N ) N p−2
|∆ak | γN
.
N =1
k=N
A(N )γN =
i=1
Z
1
α(t)dt ⇒ γN =
A(i) =
1/(N +1)
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We estimate the quantity γN :
N
X
JJ
b2 (N )
.
A(N )
Consequently
(2.2)
∞
X
I2 ≤ C
A(N )N
1− p2
r(
∞
X
)
N =1
!r |∆ak |
k=N
b2 (N )
A(N )
pr
.
By (2.1) and (2.2) we get
!r (
pr pr )
∞
∞
n or
X
X
2
b1 (N )
b2 (N )
f (m)
≤C
A(N )N r(1− p )
|∆ak |
+
.
p,r,α
A(N )
A(N )
N =1
k=N
Generalization of Lipschitz’s
Classes
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
Finally, according to Lemma 1.6, for r ≥ p we have
(
(m)
f ≤C
p,r,α
∞
X
A(N )N
r (1− p2 )
N =1
∞
X
!r |∆ak |
k=N
b(N )
A(N )
.
Now let r/p < 1. Then, according to Lemma 1.6, we have
I1 ≤ C
∞
X
A(N )N −mr
N
X
2r
n(m+1)r− p
n=1
N =1
∞
X
!r
|∆ak |
.
k=n
If we change the order of summation we get
I1 ≤ C
(2.3)
≤C
∞
X
n=1
∞
X
n=1
n
(m+1)r− 2r
p
2
nr(1− p )
∞
X
k=n
∞
X
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!r
|∆ak |
∞
X
N =n
!r
|∆ak |
k=n
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pr ) r1
b1 (n).
A(N )N −mr
Close
Now we estimate I2 . Using Lemma 1.6 and changing the order of summation we
have:
!r
∞
∞
∞
X
X
X
r(1− p2 )
I ≤C
A(N )
n
|∆a |
2
k
=C
(2.4)
=C
N =1
∞
X
n=1
∞
X
n=N
n
r (1− p2 )
∞
X
1− p2
k=n
∞
X
nr (
)
n=1
!k=n
r n
X
|∆ak |
A(N )
Generalization of Lipschitz’s
Classes
N =1
!r
|∆ak |
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
b2 (n).
k=n
From (2.3) and (2.4) we deduce
( ∞
X r
(m)
f ≤C
n
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1− p2
p,r,α
n=1
∞
X
∆ak ) r1
!r
b(n)
Contents
,
k=n
which fully demonstrates Theorem 2.1.
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Theorem 2.2. Let m be any natural number and
1 < p ≤ 2,
1 ≤ r < +∞,
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1/p + 1/q = 1.
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If an are the coefficients of series (1.1) or (1.2), then:
Close
1. For r ≤ q we have
(m)
f ≥C
p,r,α
(∞
X
n=1
n−mr |an |r b1 (n)
b1 (n)
A(n)
rq −1 ) r1
;
2. For r > q we have
(m)
f ≥C
p,r,α
(∞
X
) r1
n−mr |an |r b1 (n)
.
n=1
Proof. Let f be an even function. If f is an odd function then the proof of the theorem is analogous to the even case. It is not difficult to see that the Fourier series of
∆m f (x, t) is

∞
P
m
m

2 2
(−1)
an cos nx sinm nt,
for m even



n=1
∆m f (x, t) ∼

∞
P

m−1

 (−1) 2 −1 2m
a sin nx sinm nt, for m odd.
Generalization of Lipschitz’s
Classes
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
Title Page
n
Contents
n=1
According to the well-known Hausdorf-Young’s theorem we find
! rq
Z 2π
pr
∞
X
≥
|an |q |sin nt|mq
,
C
|∆m f (x, t)|p dx
0
n=1
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and then
∞ Z
or
n X
(m)
f ≥C
p,r,α
ν=1
1/ν
α(t)
1/(ν+1)
ν
X
! rq
q
mq
|an | |sin nt|
n=1
Using the well-known inequality sin B ≥ π2 B for 0 ≤ B ≤ π2 , we get
! rq
∞
ν
n or
X
X
(m)
q
f ≥C
A(ν)
n−mq |an |
.
p,r,α
ν=1
n=1
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dt.
Close
Let r ≤ q, then according to Lemma 1.4 we have
∞
n or
X
r
f (m)
≥
C
A(ν) ν −mq |aν |q βν q .
p,r,α
ν=1
It is easy to prove that βν ≥
(2.5)
b1 (ν)
,
A(ν)
from which we get
rq −1
∞
n or
X
b
(ν)
(m)
1
r
−mr
f ≥C
.
ν
|aν | b1 (ν)
p,r,α
A(ν)
ν=1
Generalization of Lipschitz’s
Classes
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
Let q < r, then according to Lemma 1.6 and with the change of the order of summation we have
∞
ν
n or
X
X
f (m)
≥C
A(ν)
n−mr |an |r
p,r,α
=C
(2.6)
≥C
ν=1
∞
X
n=1
∞
X
Contents
n=1
n
−mr
r
|an |
∞
X
A(ν)
ν=n
n−mr |an |r b1 (n).
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n=1
Relations (2.5) and (2.6) prove Theorem 2.2.
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We can deduce three corollaries from Theorem 2.1 and Theorem 2.2.
Corollary 2.3. Under the conditions of Theorem 2.1 and with b(n) ≤ CA(n), we
have
( ∞
!r
) r1
∞
X r 1− 2 X
(m)
f ∆ak b(n) .
p
≤C
n
p,r,α
n=1
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k=n
Close
Corollary 2.4. Under the conditions of Theorem 2.2 and with b1 (n) ≤ CA(n), we
have
(∞
) r1
X
(m)
f ≥C
n−mr |an |r b1 (n)
.
p,r,α
n=1
As a special case, for α(t) = t−βr−1 , it is easy to prove the estimates:
A(n) ≤ Cn
βr−1
and
Generalization of Lipschitz’s
Classes
βr
b(n) ≤ Cn .
Xh. Z. Krasniqi
From Theorem 2.1 and the last estimates we can deduce the following result
proved in [1].
Corollary 2.5 ([1]). Let m be any natural number and
0 < β ≤ m,
vol. 9, iss. 3, art. 73, 2008
Title Page
1 ≤ r < +∞, 1/p + 1/q = 1.
P
If the coefficients of series (1.1) or (1.2) satisfy ∞
k=1 |∆ak | < +∞, then
Contents
1 < p < +∞,
(
(m)
f ≤C
β,p,r
∞
X
n=1
1
nr(β+ q )−1
∞
X
∆ak k=n
!r ) r1
.
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References
[1] T.Sh. TEVZADZE, Some classes of functions and trigonometric Fourier series,
Some Questions of Function Theory, v. II, 31–92, Tbilisi University Press, 1981
(in Russian).
[2] M.K. POTAPOV AND M. BERISHA, Moduli of smoothnes and Fourier coefficients of functions of one variable, Publ. Inst. Math. (Beograd) (N.S.), 26(40)
(1979), 215–228 (in Russian).
[3] B. HARDY, E. LITLEWOOD
1948, 1–456 (in Russian).
AND
G. POLYA, Inequalities, GIIL Moscow,
[4] M.K. POTAPOV, A certain imbedding theorem, Mathematica (Cluj), 14(37)
(1972), 123–146.
Generalization of Lipschitz’s
Classes
Xh. Z. Krasniqi
vol. 9, iss. 3, art. 73, 2008
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