Volume 9 (2008), Issue 3, Article 73, 7 pp.
ON A GENERALIZATION OF LIPSCHITZ’S CLASSES
XH. Z. KRASNIQI
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCES ,
AVENUE "M OTHER T ERESA " 5,
P RISHTINË , 10000, R EPUBLIC OF KOSOVO
xheki00@hotmail.com
Received 26 March, 2008; accepted 06 August, 2008
Communicated by H. Bor
A BSTRACT. 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.
Key words and phrases: Lipschitz classes, Fourier series.
2000 Mathematics Subject Classification. 42A20, 42A32.
1. D EFINITIONS AND U SEFUL S TATEMENTS
We consider the series
∞
a0 X
+
an cos nx
2
n=1
(1.1)
or
∞
X
(1.2)
an sin nx
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 | < +∞
n=1
k=n
where ∆ak = ak − ak+1 (see [1]).
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
(1) α(t) ≥ C1,α , for all t ∈ [0, 1];
096-08
2
X H . Z. K RASNIQI
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
m
In [1], the classes Λ (β, p, r, ) are defined in the following way
Definition 1.2. f ∈ Λm (β, p, r, ), if
(Z Z
1
(m)
f β,p,r ≡
0
2π
|∆m f (x, t)|p
dx
tβp
0
pr
dt
t
) r1
< +∞,
where 1 < p < +∞, 1 ≤ r < +∞, β > 0, m ∈ N and
m
X
i
∆m f (x, t) =
(−1)i Cm
f [x + (m − 2i)t].
i=1
Now we define classes Λm (p, r, α) as follows:
Definition 1.3. We say f ∈ Λm (p, r, α), if
(Z
Z
f
(m)
p,r,α
1
≡
2π
α(t)
0
|∆m f (x, t)|p dx
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.
P
Lemma 1.1 ([2]). Let aν , bν and βn be numbers such that aν ≥ 0, bν ≥ 0 and ∞
ν=n aν = 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
ν=1
(2) For 1 ≤ p < ∞ we have
∞
X
aν
ν=1
ν
X
!p
≤ pp
bµ
µ=1
∞
X
aν (bν βν )p .
ν=1
Lemma 1.2 ([2]). Let aν , bν and γn be numbers such that aν ≥ 0, bν ≥ 0 and
(1) For 0 < p ≤ 1, we have
!p
∞
∞
∞
X
X
X
p
aν
bµ
≥p
aν (bν γν )p ;
ν=1
µ=ν
Pn
ν=1
aν = bn γn :
ν=1
(2) For 1 ≤ p < ∞, we have
∞
X
ν=1
aν
∞
X
µ=ν
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 73, 7 pp.
!p
bµ
≤p
p
∞
X
aν (bν γν )p .
ν=1
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G ENERALIZATION OF L IPSCHITZ ’ S C LASSES
3
Lemma 1.3 ([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 1.4 ([1]). If f ∈ W Ap , 1 < p < +∞, then
ωp(m) (h; f )
p
≤ Chmp
X
n≤[
1
h
∞
X
n(m+1)p−2
!p
|∆ak |
+C
k=n
]
X
1
h
n>[
np−2
∞
X
!p
|∆ak |
,
k=n
]
(m)
where ωp (h; f ) is the integral modulus of smoothness of order m.
2. M AIN R ESULTS
Let us denote
1/n
Z
A(n) :=
α(t)dt,
1/(n+1)
b(n) := b1 (n) + b2 (n) = n
mr
Z
1/n
mr
α(t)t
Z
1
dt +
α(t)dt.
0
1/(n+1)
We have the following first main result.
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) For p ≤ r we have
(∞
!r
pr −1 ) r1
X
∞
X
2
b(n)
(m)
f p,r,α ≤ C
nr 1− p
∆ak
b(n)
;
A(n)
n=1
k=n
1 < p < +∞,
(2) For p > r we have
(
f
(m)
p,r,α
≤C
∞
X
n
r 1− p2
n=1
∞
X
) r1
!r
∆ak
b(n)
.
k=n
Proof. Using the characteristics of the integral modulus of smoothness we have
Z 2π
pr
n
or Z 1
(m)
p
f p,r,α =
α(t)
|∆m f (x, t)| dx dt
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
1/N
α(t)dt
1/(N +1)
r
A(N ) ωp(m) (f ; 1/N ) .
N =1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 73, 7 pp.
http://jipam.vu.edu.au/
4
X H . Z. K RASNIQI
According to the Theorem 1.4, we have
n
(m)
p,r,α
f
or
≤C
∞
X
A(N )N −mr
( N
X
∞
X
n(m+1)p−2
n=1
N =1
+C
∞
X
|∆ak |
k=n
(
A(N )
N =1
!p ) pr
∞
X
∞
X
np−2
n=N +1
!p ) pr
|∆ak |
k=n
= I1 + I2 .
Now we estimate I1 and I2 . Let r/p ≥ 1. Then according to Lemma 1.1 we have
(
!p ) pr
∞
∞
X
X
|∆ak | βN
I1 ≤ C
.
A(N )N −mr N (m+1)p−2
N =1
k=N
Now we estimate the quantity βN :
A(N )N −mr βN =
∞
X
A(i)i−mr
i=N
=
mr
∞ X
i+1
i
i=N
mr
1
·
(i + 1)mr
Z
Z
1/i
α(t)dt
1/(i+1)
1/N
α(t)tmr dt,
≤2
0
or
βN ≤ C
b1 (N )
.
A(N )
Consequently
I1 ≤ C
(2.1)
∞
X
A(N )N
1− p2
r(
∞
X
)
N =1
!r |∆ak |
k=N
According to Lemma 1.2, for r/p ≥ 1 we have
(
∞
X
I2 ≤ C
A(N ) N p−2
N =1
∞
X
b1 (N )
A(N )
.
) pr
!p
|∆ak |
pr
γN
.
k=N
We estimate the quantity γN :
A(N )γN =
N
X
Z
1
α(t)dt ⇒ γN =
A(i) =
1/(N +1)
i=1
b2 (N )
.
A(N )
Consequently
I2 ≤ C
(2.2)
∞
X
A(N )N
1− p2
r(
)
N =1
∞
X
!r |∆ak |
k=N
b2 (N )
A(N )
pr
.
By (2.1) and (2.2) we get
n
f
(m)
p,r,α
or
≤C
∞
X
A(N )N
r (1− p2 )
N =1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 73, 7 pp.
∞
X
k=N
!r (
|∆ak |
b1 (N )
A(N )
pr
+
b2 (N )
A(N )
pr )
.
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G ENERALIZATION OF L IPSCHITZ ’ S C LASSES
5
Finally, according to Lemma 1.3, for r ≥ p we have
( ∞
!r pr ) r1
∞
X
X
2
b(N
)
(m)
f p,r,α ≤ C
A(N )N r(1− p )
|∆ak |
.
A(N
)
N =1
k=N
Now let r/p < 1. Then, according to Lemma 1.3, we have
I1 ≤ C
∞
X
A(N )N −mr
N
X
n
∞
X
(m+1)r− 2r
p
n=1
N =1
!r
|∆ak |
.
k=n
If we change the order of summation we get
I1 ≤ C
∞
X
n
n=1
(2.3)
≤C
∞
X
∞
X
(m+1)r− 2r
p
2
nr(1− p )
n=1
!r
|∆ak |
k=n
∞
X
∞
X
A(N )N −mr
N =n
!r
|∆ak |
b1 (n).
k=n
Now we estimate I2 . Using Lemma 1.3 and changing the order of summation we have:
!r
∞
∞
∞
X
X
X
2
I ≤C
A(N )
nr(1− p )
|∆a |
2
k
N =1
=C
∞
X
n=N
n
r (1− p2 )
∞
X
r (1− p2 )
k=n
∞
X
n=1
=C
(2.4)
∞
X
n
n=1
k=n
!r n
X
|∆ak |
A(N )
N =1
!r
|∆ak |
b2 (n).
k=n
From (2.3) and (2.4) we deduce
(
f
(m)
p,r,α
≤C
∞
X
nr
1− p2
n=1
∞
X
) r1
!r
∆ak
b(n)
,
k=n
which fully demonstrates Theorem 2.1.
Theorem 2.2. Let m be any natural number and
1 < p ≤ 2,
1 ≤ r < +∞,
1/p + 1/q = 1.
If an are the coefficients of series (1.1) or (1.2), then:
(1) For r ≤ q we have
(∞
rq −1 ) r1
X
b
(n)
(m)
1
;
f p,r,α ≥ C
n−mr |an |r b1 (n)
A(n)
n=1
(2) For r > q we have
f
(m)
p,r,α
≥C
(∞
X
) r1
n−mr |an |r b1 (n)
.
n=1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 73, 7 pp.
http://jipam.vu.edu.au/
6
X H . Z. K RASNIQI
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

∞
P

m−1

 (−1) 2 −1 2m
an sin nx sinm nt,
for m odd.
n=1
∆m f (x, t) ∼
n=1
According to the well-known Hausdorf-Young’s theorem we find
! rq
Z 2π
pr
∞
X
p
q
mq
,
C
|∆m f (x, t)| dx
≥
|an | |sin nt|
0
n=1
and then
n
f
(m)
p,r,α
or
≥C
∞ Z
X
ν=1
ν
X
1/ν
α(t)
1/(ν+1)
! rq
q
mq
|an | |sin nt|
dt.
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 p,r,α ≥ C
A(ν)
n−mq |an |
.
ν=1
n=1
Let r ≤ q, then according to Lemma 1.1 we have
∞
n
or
X
r
(m)
f p,r,α ≥ C
A(ν) ν −mq |aν |q βν q .
ν=1
It is easy to prove that βν ≥
(2.5)
n
f
b1 (ν)
,
A(ν)
(m)
p,r,α
or
from which we get
≥C
∞
X
ν
−mr
ν=1
b1 (ν)
|aν | b1 (ν)
A(ν)
r
rq −1
.
Let q < r, then according to Lemma 1.3 and with the change of the order of summation we
have
∞
ν
n
or
X
X
(m)
f p,r,α ≥ C
A(ν)
n−mr |an |r
ν=1
=C
(2.6)
≥C
∞
X
n=1
∞
X
n=1
n
−mr
r
|an |
∞
X
A(ν)
ν=n
n−mr |an |r b1 (n).
n=1
Relations (2.5) and (2.6) prove Theorem 2.2.
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)
p
f p,r,α ≤ C
n
∆ak
b(n) .
n=1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 73, 7 pp.
k=n
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G ENERALIZATION OF L IPSCHITZ ’ S C LASSES
7
Corollary 2.4. Under the conditions of Theorem 2.2 and with b1 (n) ≤ CA(n), we have
) r1
(∞
X
(m)
f p,r,α ≥ C
n−mr |an |r b1 (n)
.
n=1
As a special case, for α(t) = t−βr−1 , it is easy to prove the estimates:
A(n) ≤ Cnβr−1
and
b(n) ≤ Cnβr .
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,
1 ≤ r < +∞, 1/p + 1/q = 1.
P
If the coefficients of series (1.1) or (1.2) satisfy ∞
k=1 |∆ak | < +∞, then
( ∞
!r ) r1
∞
X
X
1
(m)
f
≤C
nr(β+ q )−1
∆a
.
1 < p < +∞,
k
β,p,r
n=1
k=n
R EFERENCES
[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
sian).
AND
G. POLYA, Inequalities, GIIL Moscow, 1948, 1–456 (in Rus-
[4] M.K. POTAPOV, A certain imbedding theorem, Mathematica (Cluj), 14(37) (1972), 123–146.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 73, 7 pp.
http://jipam.vu.edu.au/