THE INTEGRAL WAVELET TRANSFORM
IN WEIGHTED SOBOLEV SPACES
NGUYEN MINH CHUONG AND TA NGOC TRI
Received 27 September 2001
The integral wavelet transform is defined in weighted Sobolev spaces, in which
some properties of the transform as well as its asymptotical behaviour for small
dilation parameter are studied.
1. Preliminaries and notations
It is well known that the integral wavelet transform is a very powerful tool to
study sciences and technology. In [5] wavelet theory has been investigated in
very much functional spaces, even in BMO, VMO (for further details of those
spaces, please refer to [5] and the references therein) even for pseudodifferential
operators, however, the integral wavelet transform in weighted Sobolev spaces
has not been studied yet neither in [5] nor in any other work.
The aim of this paper is to study this unsolved problem.
Let ωµ (x) ∈ L∞ (Rn ), ωµ (x) > 0, for almost all x ∈ Rn and for each x,
ωµ (x + y) ≤ C1,µ ωµ (x),
(1.1)
for almost all y ∈ Rn , where µ is a multi-index.
We use the Sobolev space with weighted norm defined as follows:
m,p Wω
Rn = f ∈ L p Rn | ∂k f ∈ L p Rn , |k | ≤ m
(1.2)
equipped with the norm
f m,p,ω =
|µ|≤m
Rn
p
ωµ (x)∂µ f (x) dx
1/ p
where µ = (µ1 , . . . , µn ), |µ| = µ1 + · · · + µn , µi ≥ 0.
Copyright © 2002 Hindawi Publishing Corporation
Abstract and Applied Analysis 7:3 (2002) 135–142
2000 Mathematics Subject Classification: 42Cxx, 42C40, 65Txx, 65T60
URL: http://dx.doi.org/10.1155/S1085337502000775
< ∞,
(1.3)
136
The integral wavelet transform in weighted Sobolev spaces
Let ᏿(Rn ) be the Schwartz space of all differentiable functions ϕ on Rn such
that for all multi-indices α and β
sup xα Dβ ϕ(x) < ∞.
(1.4)
x∈Rn
The Fourier transform Ᏺ : f → fˆ is given by
fˆ(y) = (2π)−n/2
Rn
f (x)e−i(x,y) dx,
(1.5)
where (x, y) = x1 y1 + · · · + xn yn , x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) (see [1, 2, 4]).
m,p
As traditionally, it is not difficult to prove that ᏿(Rn ) is dense in Wω (Rn ).
1
n
Now we recall that a basic wavelet is a nontrivial function ψ ∈ L (R ) such that
its integral on Rn is 0 and its Fourier transform ψ̂(ξ) satisfies the condition
(2π)
n
∞ 2
ψ̂(aξ)
da,
a
0
(1.6)
denoted by Cψ which is a constant for every ξ = 0 and Cψ = 0.
With a basic wavelet ψ and a function f ∈ ᏿(Rn ), we define the following integral:
1
1
Lψ f (b, a) = 2n
|
Cψ a|n
Rn
ψ̄
t−b
f (t) dt,
a
(1.7)
where b ∈ Rn and a ∈ R \ {0} (see [3, 5, 6]).
2. Some properties
Proposition 2.1. If f ∈ ᏿(Rn ) then
Lψ f (·, a)
m,p,ω
≤ C f m,p,ω ,
(2.1)
where a ∈ R, a = 0 and fixed, C is a constant independent of f .
Proof. Obviously we have
1 Lψ f (·, a) = f ∗ D−a ψ̄ (·),
2n
Cψ
(2.2)
where Da : L2 (Rn ) → L2 (Rn ) and (Da ψ)(a) = |a|−n/2 ψ(x/a); a = 0.
Since f ∈ ᏿(Rn ) the differentiation and integration can be interchanged
1 −a
D ψ̄ ∗ ∂µ f (·).
∂µ Lψ f (·, a) = 2n
Cψ
(2.3)
N. M. Chuong and T. N. Tri
137
It is not difficult to see that
Lψ f (·, a)
m,p,ω
=
Rn
|µ|≤m
=
2n
2n
1
≤ ωµ (x)
n
|µ|≤m R
Cψ
n
|µ|≤m R
Cψ
p
Rn
Cψ
1
2n
1
|µ|≤m
≤ ωµ (·)∂µ Lψ f (·, a) d(·)
Rn
Rn
1/ p
p
∂µ f (x − y) D−a ψ̄ (y) d y dx
µ ∂ f (x− y) p D−a ψ̄ (y) p ωµ (x) dx
−a
D ψ̄(y)
p
ωµ (x) ∂µ f (x − y) dx
Rn
1/ p
1/ p
dy
1/ p
d y.
(2.4)
However,
Rn
p
ωµ (x) ∂µ f (x − y) dx =
Rn
p
p
ωµ (u + y) ∂µ f (u) du
≤ C1,µ
Rn
ωµ (u) ∂µ f (u) du.
(2.5)
Consequently,
p
ωµ (·)∂µ Lψ f (·, a) d(·)
Rn
≤
C1,µ
2n
1/ p Cψ
≤ Cµ
Rn
−a
D ψ̄(y) d y
Rn
1/ p
Rn
p
ωµ (x) ∂µ f (x) dx
p
ωµ (x) ∂µ f (x) dx
1/ p
(2.6)
1/ p
,
where
Cµ =
C1,µ
1/ p
|a|n/2 ψ 1 .
2n
Cψ
(2.7)
Therefore,
Lψ f (·, a)
m,p,ω ≤ C
|µ|≤m
Rn
p
ωµ (x) ∂µ f (x) dx
1/ p
,
(2.8)
where
C = max Cµ ,
|µ|≤m
(2.9)
138
The integral wavelet transform in weighted Sobolev spaces
that is,
Lψ f (·, a)
m,p,ω
≤ C f m,p,ω .
(2.10)
By Proposition 2.1, we extend (Lψ f (·, a)) for fixed a to a continuous mapping
m,p
from Wω (Rn ) to itself. It is called the integral wavelet transform in weighted
Sobolev space.
m,p
Theorem 2.2. If ψ and ϕ are basic wavelets and f , g belong to Wω (Rn ), then the
following estimate holds true:
Lψ f (·, a) − Lϕ g (·, a)
m,p,ω


ψ
ϕ
ϕ
1


≤ C 1 |a|n/2  − f m,p,ω + f − g m,p,ω  ,
2n
2n
2n C
Cϕ Cϕ
ψ
(2.11)
1
where C 1 is a constant independent of f and g.
Proof. It is sufficient to prove the case f , g ∈ ᏿(Rn ).
Obviously
Rn
p
ωµ (·)∂µ Lψ f − Lϕ g (·, a) d(·)



=
ωµ (·) ∂µ f

n
 R
1/ p
1/ p
p


 D ψ̄ D ϕ̄  ∗  −  (·) d(·)
2n
2n


Cψ
Cϕ


−a
−a



p 1/ p




−a
−a
ψ̄
D
D
ϕ̄


µ
=
ωµ (x) ∂ f (x − y)  −  (y) d y dx
2n
2n
Rn


 Rn

Cψ
Cϕ

 p 1/ p
−a
−a
µ  D ψ̄ D ϕ̄  ≤
ωµ (x) ∂ f (x − y) dx
dy
 −  (y)
2n
2n
n
n
R R
Cψ
Cϕ
≤ C2,µ
where
Rn
p
ωµ (x) ∂µ f (x) dx
1/ p
,
(2.12)
ψ
1/ p
ϕ n/2 .
C2,µ = |a| − C1,µ
2n
2n C
Cϕ ψ
1
(2.13)
N. M. Chuong and T. N. Tri
So
139
ψ
ϕ 1 n/2 − f m,p,ω , (2.14)
m,p,ω ≤ C |a|
2n
2n C
Cϕ ψ
Lψ f (·, a) − Lϕ f (·, a)
1
where
C 1 = max C1,µ
1/ p
|µ|≤m
.
(2.15)
Similarly we obtain
Lϕ f (·, a) − Lϕ g (·, a)
ϕ1
≤ C 1 |a|n/2 f − g m,p,ω .
m,p,ω
2n
(2.16)
Cϕ
By the triangle inequality we get
Lψ f (·, a) − Lϕ g (·, a)
m,p,ω


ψ
ϕ ϕ1


≤ C 1 |a|n/2  − f m,p,ω + f − g m,p,ω  .
2n
2n
2n C
Cϕ Cϕ
ψ
(2.17)
1
3. Symptotical behaviour for small dilation parameter
From Theorem 2.2 the following proposition follows immediately.
m,p
Proposition 3.1. If ψ is a basic wavelet and f ∈ Wω (Rn ), then
Lψ f (·, a)
m,p,ω
= O |a|n/2 .
(3.1)
Now consider the operator
Λψ f (b, a) = ψa ∗ f (b) =
1
an
Rn
f (t)ψ
b−t
dt,
a
(3.2)
where ψ ∈ L1 (Rn ), f ∈ L p (Rn ), 1 ≤ p < ∞, and
1
x
,
ψa (x) = n ψ
a
a
a = 0.
(3.3)
In the sequel, the following lemma is needed.
m,p
Lemma 3.2. Let f ∈ Wω (Rn ), 1 ≤ p < ∞, ψ ∈ L1 (Rn ) ∩ Lq (Rn ), with
Rn
ψ(t) dt = 1,
1 1
+ = 1.
p q
(3.4)
140
The integral wavelet transform in weighted Sobolev spaces
Then
m,p
(i) (Λψ f )(·, a) → f (·) in Wω (Rn ) as a → 0+ ;
µ
(ii) [∂ (Λψ f )](·, a) = [Λψ (∂µ f )](·, a) = a−|µ| (Λ∂µ ψ f )(·, a) if ∂µ ψ ∈ L1 (Rn )
∩ Lq (Rn ), |µ| < m, a > 0.
Proof. Since (Λψ f )(·, a) = (ψa ∗ f )(·), and for each multi-index α,
such that ∂µ f ∈ L p (Rn ), ψ ∈ Lq (Rn ), we obtain
µ
∂ ψa ∗ f
n
(·) = ψa ∗ ∂µ f (·),
Λψ f (·, a) − f (·)
m,p,ω
1/ p
p
=
ωµ (·)∂µ Λψ f (·, a) − f (·) d(·)
|µ|≤m
=
Rn
Rn
|µ|≤m
≤m
i=1 αi
p
ωµ (·)Λψ ∂µ f (·, a) − ∂µ f (·) d(·)
(3.5)
1/ p
.
Taking into account that ∂µ f ∈ L p (Rn ), it is easy to see that
Rn
p
ωµ (·)Λψ ∂µ f (·, a) − ∂µ f (·) d(·)
≤ ω µ ∞
Rn
µ Λ ∂ f (·, a) − ∂µ f (·) p d(·) −→ 0
(3.6)
as a −→ 0+ .
m,p
Consequently, (Λψ f )(·, a) → f (·) in Wω (Rn ) as a → 0+ , that is, (i) is proved.
m,p
To check (ii) take fr ∈ ᏿(Rn ), fr → f in Wω (Rn ). It is obvious that in
m−|µ|,p
(Rn )
Wω
µ
∂ Λψ fr (·, a) = Λψ ∂µ fr (·, a) = a−|µ| Λ∂µ ψ fr (·, a).
(3.7)
By the continuity of the operators
m,p m,p Rn −→ Wω Rn ,
m−|µ|,p n m,p R ,
∂µ : Wω Rn −→ Wω
Λψ : Wω
(3.8)
for ψ ∈ Lq (Rn ), |µ| < m, letting r → ∞ we get (ii).
m,p
Theorem 3.3. Let f ∈ Wω (Rn ), 1 ≤ p < ∞, ψ ∈ L1 (Rn ) ∩ Lq (Rn ) such that
Rn
ψ(t) dt = 1.
(3.9)
Moreover, assume that ∂µ ψ is a basic wavelet for each multi-index µ, |µ| ∈
{0, 1, 2, . . ., m}, and ∂µ ψ ∈ Lq (Rn ), with 1/ p + 1/q = 1. Suppose additionally that f
and ψ are real-valued functions and a > 0. Then
1 1
lim+ |µ|+n/2 L∂µ ψ f (·, −a) − 2n
a→0 a
C
∂µ ψ
∂ f (· )
µ
m−|µ|,p,ω
= 0.
(3.10)
N. M. Chuong and T. N. Tri
141
Proof. Obviously
an/2 µ L∂µ ψ f (·, −a) = ∂ ψ a ∗ f (· )
2n
C∂µ ψ
an/2 = 2n
C∂µ ψ
a|µ|+n/2 µ ψa ∗ ∂ µ f ( · ) = ∂ ψa ∗ f (·).
2n
C∂µ ψ
(3.11)
Under the assumptions of the theorem, the differentiation and integration can
be interchanged, and furthermore by the continuity of the operator
m−|µ|,p Rn ,
∂µ : Wω m,p Rn −→ Wω
(3.12)
for |µ| < m, we get
1 1
µ
∂ f (· )
|µ|+n/2 L∂µ ψ f (·, −a) − 2n
a
C∂µ ψ
≤ 2n
1
C∂µ ψ
C
≤ 2n
C∂µ ψ
m−|µ|,p,ω
µ ∂ ψa ∗ f ( · ) − ∂ µ f ( · ) m−|µ|,p,ω
(3.13)
ψa ∗ f ( · ) − f ( · ) m,p,ω .
Lemma 3.2 implies now that the last term in (3.13) tends to 0 as a → 0+ .
Acknowledgment
This paper was supported by the National Fundamental Research Fund Grant
for Natural Science, Vietnam.
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[1]
[2]
[3]
[4]
[5]
[6]
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N. M. Chuong, N. M. Tri, and L. Q. Trung, Theory of Partial Differential Equations,
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L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution
Theory and Fourier Analysis, Fundamental Principles of Mathematical Sciences,
vol. 256, Springer-Verlag, Berlin, 1983.
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142
The integral wavelet transform in weighted Sobolev spaces
Nguyen Minh Chuong: Institute of Mathematics, P.O. Box 631, Bo Ho, 10 000
Hanoi, Vietnam
E-mail address: nmchuong@thevinh.ncst.ac.vn
Ta Ngoc Tri: University of Pedagogy Hanoi II, Me Linh, Vinh Phu, Vietnam