Gen. Math. Notes, Vol. 14, No. 1, January 2013, pp.1-5
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
On Estimates for the Fourier Transform
in the Space Lp(R)
M. El Hamma1 and R. Daher2
1,2
Faculty of Sciences Aı̈n Chock, University of Hassan II, Casablanca, Morocco
1
E-mail: m elhamma@yahoo.fr
2
E-mail: ra daher@yahoo.fr
(Received: 1-8-12 / Accepted: 18-11-12)
Abstract
Using a Steklov’s function, we obtain two estimates are proved in certain
classes of functions characterized by a generalized continuity modulus.
Keywords: Fourier transform; Steklov function; generalized continuity
modulus.
1
Introduction and Preliminaries
It is well known that Fourier transforms are widely used in mathematical
physics. Certain applications of this transform are described in a number
of fundamental monographs (e.g., see [2, 3, 6, 7]). In this paper, we prove
two estimates in certain classes of functions characterized by a generalized
continuity modulus and connected with the Fourier transform in the space
Lp (R), 1 < p ≤ 2, analogs of the statements proved in [1].
Assume that Lp (R), (1 < p ≤ 2), is the space of p-power integrable functions f : R −→ C with the norm
kf kp =
Z
|f (x)|p dx
1/p
.
R
It is well known that the Fourier transform of a function f ∈ L1 (R) is
defined by
2
M. El Hamma et al.
1 Z
fb(x) = √
f (t)e−ixt dt.
2π R
The inverse Fourier transform is defined by
1 Z b
f (t) = √
f (x)eixt dx.
2π R
We have the Hausdroff-Young inequality
kfbkq ≤ Ckf kp ,
(1)
where p1 + 1q = 1 and C is a positive constant.
In Lp (R), consider Steklov’s function
1 Z x+h
f (t)dt, h > 0.
2h x−h
We define the differences of first and higher orders as follows:
Fh f (x) =
∆h f (x) = Fh f (x) − f (x) = (Fh − I)f (x),
∆kh f (x) = ∆h (∆hk−1 f (x)) = (Fh − I)k f (x) =
k
X
(−1)k−i (ki )Fih f (x),
i=0
where F0h f (x) = f (x), Fih f (x) = Fh (Fi−1
h f (x)), i = 1, 2, .., k; k = 1, 2, ... and
p
I is the unit operator in the space L (R).
The quantity
Ωk (f ; δ) = sup k∆kh f (x)kp
0<h≤δ
is called the generalized continuity modulus of k th order of the function f ∈
Lp (R).
r,k
Denote by Wp,φ
(R) the class of functions f ∈ Lp (R) having the generalized
derivatives f 0 (x), f 00 (x), ..., f (r) (x) in the sence of Levi (see [4, 5]) in Lp (R) such
that
Ωk (f (r) , δ) = O(φ(δ k )),
where φ(t) is a continuous steadily increasing function on [0, +∞) and φ(0) =
0.
In [1], we have
3
On Estimates for the Fourier Transform...
1 Z sin(ht) b ixt
√
Fh f (x) =
f (t)e dt,
2π R ht
then
1 Z sin(ht)
Fh f (x) − f (x) = √
− 1)fb(t)eixt dt
(
ht
2π R
By Hausdroff-Young inequality (1), we obtain
!1/q
sin(ht)
|
− 1|q |fb(t)|q dt
ht
R
Z
≤ CkFh f (x) − f (x)kp .
r,k
Hence, for any function f ∈ Wp,φ
(R), we have
!1/q
sin(ht) qk b q
t |1 −
| |f (t)| dt
ht
R
Z
2
qr
≤ Ck∆kh f (r) (x)kp .
(2)
Main Result
In this section, we estimate the integral
Z
|fb(x)|q dx
|x|≥N
in certain classes of functions in Lp (R).
Theorem 2.1 It holds that
!
Z
sup
q
|fb(t)|
dt = O N
|t|≥N
r,k
f ∈Wp,φ
(R)
2
φ (( )k ) ,
N
−qr q
where r = 0, 1, ...; k = 1, 2, ....; and φ(t) is any nonnegative function defined
on the interval [0, ∞).
r,k
Proof. Let f ∈ Wp,φ
(R). By the Hölder inequality, we have
Z
|fb(t)|q dt −
|t|≥N
=
Z
|t|≥N
(1 −
|t|≥N
≤
Z
Z
|t|≥N
Z
sin(ht) b q
sin(ht) b q
|f (t)| dt =
(1 −
)|f (t)| dt
ht
ht
|t|≥N
sin(ht) b q− 1 b 1
)|f (t)| k |f (t)| k dt
ht
! qk−1
qk
q
|fb(t)| dt
!
sin(ht) qk b q
|1 −
| |f (t)| dt
ht
|t|≥N
Z
1
qk
4
M. El Hamma et al.
! qk−1
qk
Z
=
q
|fb(t)| dt
|t|≥N
≤ N
Z
!
−qr
t
|t|≥N
! qk−1
qk
q
b
|f (t)| dt
Z
−r/k
|t|≥N
sin(ht) qk b q qr
| |f (t)| t dt
|1 −
ht
1
qk
!
sin(ht) qk b q
| |f (t)| dt
t |1 −
ht
|t|≥N
Z
1
qk
qr
From formula (2), we conclude that
Z
tqr |1 −
|t|≥N
sin(ht) qr b q
| |f (t)| dt ≤ C q k∆kh f (r) (x)kqp .
ht
Therefore
Z
|fb(t)|q dt ≤
|t|≥N
! qk−1
Z
qk
sin(ht) b q
|f (t)| dt+C q N −r/k
|fb(t)|q dt
k∆kh f (r) (x)k1/k
p .
ht
|t|≥N
Z
|t|≥N
Note that
Z
|t|≥N
sin(ht) b q Z
sin(ht) b q
1 Z
|
|fb(t)|q dt
|f (t)| dt ≤
||f (t)| dt ≤
ht
ht
N h |t|≥N
|t|≥N
Then
! qk−1
Z
Z
qk
1
q
q
−r/k
q
|fb(t)| dt ≤
|fb(t)| dt+C N
|fb(t)| dt
k∆kh f (r) (x)k1/k
p .
N h |t|≥N
|t|≥N
|t|≥N
Z
q
Setting h =
2
h
we obtain
! qk−1
Z
qk
1Z
q
q −r/k
q
b
b
|f (t)| dt ≤ C N
|f (t)| dt
k∆kh f (r) (x)k1/k
p .
2 |t|≥N
|t|≥N
Since
k∆kh f (r) (x)kp = O[φ(
2 k
) ].
N
Consequently
Z
|t|≥N
which completes the proof.
|fb(t)|q dt = O[N −qr φq ((
2 k
) )]
N
On Estimates for the Fourier Transform...
5
r,k
Corollary 2.2 Let φ(t) = tα , α > 0, and let f ∈ Wp,t
α (R). Then
Z
|fb(t)|q dt = O(N −qr−qkα )
|t|≥N
r,k
Proof. Let f ∈ Wp,φ
(R) and φ(t) = tα , α > 0. Then by Theorem 2.1 we
have
Z
|fb(t)|q dt = O(N −qr N −qkα )
|t|≥N
= O(N −qr−qkα )
which completes the proof.
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