WAVELETS IN WEIGHTED SOBOLEV SPACE
1,*
Pandey, C. P.,
1
Department of Mathematics,
2Department of Mathematics
3Department of Mathematics
ARTICLE INFO ABSTRACT
Wavelet transform is studied on Sobolev space is obtained. Wavelet tra
nsform with compactly supported wavelet is studied. Asymptotic will also be discussed.
Copyright © Pandey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, di and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
Let
L
2(
R) be the analyzing wavelet and
(
x
)
(
x
b
),
b
T
b Rand the dilation operator Da by
,
a
a
x
|
a
|
)
x
(
D
a 1/2 R0 = R\{0}.and a unitary transformation U(b,a) : L2(R, dt)
(x)
T
D
(
x
)
|
a
|
)
a
,
b
(
U
b a 1/2The actions of the Fourier transform
R
x i
,
dx
e
)
x
(
f
)
(
f
)
)(
Ff
(
on the operators Tb and Da are given by
,
F
e
FT
b(.)b
.
F
D
FD
a
1/a*Corresponding author: Pandey, C. P., Department of Mathematics
ISSN: 0975-833X
Article History:
Received 28th September, 2013 Received in revised form 21st October, 2013
Accepted 17th December, 2013 Published online 26th January,2014
Key words:
Sobolev space, Weighted Sobolev space, Wavelet transforms.
RESEARCH ARTICLE
WAVELETS IN WEIGHTED SOBOLEV SPACE
,
2Rakesh Mohan and
3Bhairaw Nath Tripathi
, Ajay Kumar Garg Engineering College, Ghaziabad 201009,
Department of Mathematics, Dehradun Institute of Technology, Dehradun 248009, India
Department of Mathematics, Uttarakhand Technical University, Dehradun 248007, India
ABSTRACT
Wavelet transform is studied on the weighted Sobolev space Sobolev space is obtained. Wavelet tra
nsform with compactly supported wavelet is studied. Asymptotic will also be discussed.
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, di edium, provided the original work is properly cited.
) be the analyzing wavelet and
f
L
2(
R) be any function. We define the translation operator T(1.1)
{0}. (1.2)
, dt) L2 (R, dt) by
,
(
b
,
a)
a
b
x
R x R0.
R (1.3)(1.4)
(1.5)
Department of Mathematics, Ajay Kumar Garg Engineering College, Ghaziabad 201009, India Available online at http://www.journalcra.com
International Journal of Current Research
Vol. 6, Issue, 01, pp.4555-4561, January,2014
INTERNATIONAL
z
Bhairaw Nath Tripathi
Engineering College, Ghaziabad 201009, India
nology, Dehradun 248009, India
Uttarakhand Technical University, Dehradun 248007, India
.
k
B
Boundedness results in thisproperties of the wavelet transform
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,
) be any function. We define the translation operator Tb by
(1.1)
Ajay Kumar Garg Engineering College, Ghaziabad 201009, India
Definition1.1. A function
L2(R,dt) is admissible only if is not identical to zero and
0
R R
2 2
0
a
dadb
,
)
a
,
b
(
U
. (1.6)Lemma1.2.
L2(R,dt)\{0} is admissible if and only if the integral
0
R
2
d
|
|
|
)
(
|
exists.
Prof. See [4, p. 877].
Lemma1.3. Let be admissible and
f
L2(R,dt) Let
0
R
2
.
d
|
|
)
(
C
The integral
0
)
a
,
b
(
U
,
f
C
1
a)
(b,
f
~
a)
(b,
)
f
L
(
(1.7)
R
dt
)
t
(
f
a
b
t
|
a
|
1
C
1
(1.8)
defines an element of L2 (RxR0,
)
a
dbda
2 .
Moreover,
:
L
L2(R,dt)L2 (RxR0,)
a
dbda
2 is an isometry.
Proof. See [4, p, 877).
From (1.5), the Fourier transform of L with respect to its translation argument is given by
)
(
f
)
a
(
|
a
|
C
1
)
(
)
a
)(.,
f
L
(
21
. (1.9)
Definition1.4. The operator
L
:
L2(R,dt)L2 (RxR0,)
a
dbda
2 is called wavelet transform with respect to analyzing wavelet
.
In this paper, we extend the wavelet transform, which we defined on L2(R, dt), to weighted Sobolev space Bk and boundedness properties will be investigated. Asymptotic properties for small dilation parameter will also be studied.
The Weighted Sobolev Space
B
k.
(1)
0
(
0
)
(
n
)
(
)
(
);
,
Rn (2.1)(2)
0
1 n
|
|
1
)
(
, (2.2)
(3)
(
)
a
b
log
(1
|
|),
Rn (2.3)for some real number a and position real number b. We denote by Mc the set of all
M satisfying condition witha concave function on
[
0
,
)
We suppose
M
c from now on.Let
M
c. We denote by S the set of all functions
L1(Rn) with the property that and
C
and for each multi index and each non-negative numberwe have
;
|
)
x
(
D
|
e
sup
)
(
p
(x)R x ,
n
(2.4)
(
)
sup
e
|
D
(
)
|
) (
R ,
n
. (2.5)
The topology of S is defined by the semi-norms p and . The dual of Sis denoted by S'the elements of which are called
ultra-distributions. We may refer to [1] for its various properties. We note that for = log (1+), S is reduces to S, the
Schwartz space.
We also recall the definition of test function space D. The space Dis the set of all in L
1
(Rn) such that has compact support and
||
||
for all > 0 and
nR
) (
e
)
(
d (2.6)Now, let
M
c.
Then K is defined to be the set of positive function k in Rn with the following property. There exists >0such that
)
e
k(
)
for
all
,
(
k
( ) Rn. (2.7)Let
M
c,
k
K
and
1
p
.
Then weighted Sobolev space B,k(Rn) is defined to be the space of all ultra-distributionsf
S
' such that;
)
(
)
(
||
||
2 1 2
n
R
k
k
f
d
f
(2.8)and
.
)
(
f
)
k(
sup
ess
||
f
||
,k
(2.9)
Note that the space Bk(Rn) is a generalization of the HÖrmander space
Bk (Rn) [2 ] and reduces to the space Bk(Rn) for= log (1+).
The Wavelet Transform on Weighted Sobolev Space Bk
.
In this section we define the space
W
k of all measurable functions f on RxR0 such that
(
,
)
,
)
,
(
2 1 2 0
R k Wa
da
a
b
f
a
b
f
k (3.1)Theorem3.1. Assume that analyzing wavelet
L
2satisfies the following admissibility condition:
Rd
C
.
|
|
2
(3.2)
Let (Lf)(b,a) be the wavelet transform of the function
f
B
k with respect to the analyzing wavelet
.
2L
Then
(
,
)
||
||
k,
W
f
a
b
f
L
k
(3.3)
Proof. From (3.1), we have
a
da
a
b
f
L
a
b
f
L
R kWk
0 2 2)
,
)(
(
,
0 2 2|
)
(
)
,
)(
(
|
|
)
(
|
R R
a
da
d
a
b
f
L
k
=
0 2 22
1
|
(
)
|
|
f
(
)
|
|
)
(
|
R R
a
da
d
a
a
C
k
R Rd
f
k
du
u
u
C
2 2|
)
(
)
(
|
|
|
|
)
(
|
1
0 k Rf
du
u
u
C
|
|
||
||
|
)
(
|
1
0 2
.
||
f
||
||
||
1
k
C
f
kC
Theorem3.2. For admissible and integrable 1, 2 and
f
,
g
B k..
||
||
)
,
(
)
,
(
2 2 1 2 1 2 1 2 2 1
k L k Lk
C
C
f
C
f
g
a
b
g
L
a
b
f
L
Proof. We have
k
k
L
f
b
a
L
f
b
a
a
b
g
L
a
b
f
L
(
,
)
(
,
)
(
,
)
(
,
)
2 1
2
1
ka
b
g
L
a
b
f
L
(
,
)
(
,
)
2
2
.(3.4)Now,
21 2 2
|
)
(
|
)
(
)
,
(
)
,
(
||
)
,
(
)
,
(
||
2 1 21
f
b
a
L
f
b
a
L
f
b
a
L
f
b
a
k
d
L
=
2 1
2 2
2 2 1
1 2 1
|
)
(
|
)
(
)
(
|
|
1
)
(
)
(
|
|
1
2
1
R
d
k
f
a
a
C
f
a
a
C
.
)
(
|
)
(
|
)
(
|
|
2 1 2
2 1
2 2
2
1
a
C
C
d
f
k
a
R
(3.5)
Now, using the inequality
1
L
||
||
)
(
,
we have
;
C
C
C
C
1 2 1
2
1 L
2 1
2 1
so that
2
2 1
2
2 1
1 2 1
2
1
C
C
C
LC
. (3.6)
Using (3.6) in (3.5), we have
k
L
k
f
C
C
a
a
b
f
L
a
b
f
L
(
,
)
(
,
)
||
|
|
||
||
||
1 2 1
2 1
2 1
2 1
. (3.7)Similarly, we can write
k
L
k
a
f
g
C
a
b
g
L
a
b
f
L
(
,
)
(
,
)
||
|
|
||
||
||
21 2
1 2 2
2
. (3.8)
Invoking (3.8), (3.4), we have
k
L
k
f
C
C
a
a
b
g
L
a
b
f
L
(
,
)
(
,
)
||
|
|
||
||
||
1 2 1
2 1
2 1
2 1
.k
L
g
f
C
||
||
1 2
2
Asymptotic Behavior for Small Dilation Parameters.
Let us recall the equation (1.9):
R.
dt
)
t
(
f
a
b
t
|
a
|
1
C
1
)
a
,
b
(
f
L
(4.1)In what follows we assume that is real valued and a>0. Let us define
.
a
x
a
1
)
x
(
a
(4.2)Let us use the notation
f
(
t
)
dt
a
t
b
a
1
)
b
(
f
)
a
,
b
(
f
Ra
. (4.3)From (4.1) and (4.3), we have
L
f
(
b
,
a
)
a
C
)
a
,
b
(
f
)
b
(
f
a
. (4.4)Theorem4.1. Let
f
B
R
1
(t)dt
with
)
(
L
and
1
R
k . Thenin
(.)
f
)
a
(.,
f
Bwkas
a
0;
Proof. In view of (4.3) we have
(i)
R a
k
a
f
f
f
f
k
d
2 2 2|
)
(
|
)
(
Ra
f
f
d
2 2|
)
k(
|
)
(
)
(
Rd
k
f
a
b
f
L
a
C
2 2 2 1)
(
)
(
)
(
,
2 2 1 2 1 2 1)
(
)
(
f
)
(
|
|
1
.
Rf
a
a
C
a
C
a
f
f
k
d
R 2 2
|
)
(
|
)
(
)
(
)
(
a
d
k
R 2 2 2)
(
1
)
(
f
)
(
a
d
k
R 2 2)
(
1
)
(
f
)
(
Rd
a
I
(
,
)
|
,
where |I (a, )| =
k
(
)
f
(
)
1
-
(
a
)
.
Then we have
lim
|
I
(
a
,
)
|
0
0
a
a.e.Let us now set
M
sup
1
(
a
)
,
R
which is independent of a.
Then
)
(
f
)
k(
M.
)
,
a
(
I
.
Now, applying the dominated convergence theorem, we have
a
f
f
(.,
a
)
f
(.)
in
Bkas
a
0
.
REFERENCES
[1]. G. BjÖrck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1905), 351-407.
[2]. L. HÖrmonder, The Analysis of Linear Partial Differential Operators I, Springer-Verlag Berlin Heidelberg (1983). [3]. R.S. Pathak, Generalized Sobolev spaces and pseudo-differential operators on spaces of ultra-distributions, in Structure of
Solutions of Differential Equations (Eds. M. Morimoto, T.Kawai), World Scientific, Singapore (1996), 343-368.
[4]. A. Rieder, The wavelet transform on Sobolev spaces and its approximation properties, Numer. Math. 58 (1991), 875-894.
