BOUNDEDNESS OF THE WAVELET TRANSFORM
IN CERTAIN FUNCTION SPACES
Wavelet Transform
R.S. PATHAK AND S.K. SINGH
Department of Mathematics,
Banaras Hindu University,
Varanasi - 221 005, India
EMail: ramshankarpathak@yahoo.co.in
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
Title Page
Contents
Received:
06 October, 2005
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Accepted:
25 January, 2007
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Communicated by:
L. Debnath
2000 AMS Sub. Class.:
42C40, 46F12.
Key words:
Continuous wavelet transform, Distributions, Sobolev space, Besov space,
Lizorkin-Triebel space.
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Abstract:
Using convolution transform theory boundedness results for the wavelet transn
form are obtained in the Triebel space-LΩ,k
p , Hörmander space-Bp,q (R ) and
general function space-L∞,k , where k denotes a weight function possessing specific properties in each case.
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Contents
1
Introduction
3
2
Boundedness of W in LΩ,k
p
4
3
Boundedness of W in Bp,k
8
Wavelet Transform
4
A General Boundedness Result
10
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
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1.
Introduction
The wavelet transform W of a function f with respect to the wavelet ψ is defined by
(1.1)
f˜(a, b) = (Wψ f )(a, b) =
Z
f (t)ψ a,b (t)dt = (f ∗ ha,0 )(b),
Rn
n
where ψ a,b = a− 2 ψ( x−b
), h(x) = ψ(−x), b ∈ Rn and a > 0, provided the integral
a
exists. In view of (1.1) the wavelet transform (Wψ f )(a, b) can be regarded as the
convolution of f and ha,0 . The existence of convolution f ∗ g has been investigated
by many authors. For this purpose Triebel [6] defined the space LΩ,k
and showed
p
Ω,k
Ω,k
that for certain weight functions k, f ∗ g ∈ Lp , where f, g ∈ Lp , 0 < p ≤
1. Convolution theory has also been developed by Hörmander in the generalized
Sobolev space Bp,q (Rn ), 1 ≤ p ≤ ∞.
In Section 2 of the paper, a definition and properties of the space LΩ,k
are given
p
and a boundedness result for the wavelet transform Wψ f is obtained. In Section 3 we
recall the definition and properties of the generalized Sobolev space Bp,q (Rn ) due
to Hörmander [1] and obtain a certain boundedness result for Wψ f . Finally, using
Young’s inequality a third boundedness result is also obtained.
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
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2.
Boundedness of W in LΩ,k
p
Let us recall the definition of the space LΩ,k
by Triebel [6].
p
Definition 2.1. Let Ω be a bounded C ∞ -domain in Rn . If k(x) is a non-negative
weight function in Rn and 0 < p ≤ ∞, then
Ω,k
(2.1) Lp = f |f ∈ S 0 , supp F f ⊂ Ω;
Z
k f kLΩ,k
=k kf kLp =
p
p
p
k (x)|f (x)| dx
p1
<∞ .
Theorem 2.1 (Hans Triebel). If k is one of the following weight functions:
k(x) = |x|α ,
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
Rn
If k(x) = 1 then LΩ,k
= LΩ
p
p.
We need the following theorem [6, p. 369] in the proof of our boundedness result.
(2.2)
Wavelet Transform
α≥0
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k(x) =
(2.3)
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|xj |αj ,
αj ≥ 0
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j=1
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(2.4)
γ
k(x) = kβ,γ (x) = eβ|x| ,
β ≥ 0, 0 ≤ γ ≤ 1
and 0 < p ≤ 1, then
(2.5)
⊂ LΩ,k
LΩ,k
∗ LΩ,k
p
p
p
and there exists a positive number C such that for all f, g ∈ LΩ,k
p ,
kf ∗ gkLkp ≤ C kf kLkp kgkLkp .
(2.6)
Using the above theorem we obtain the following boundedness result for the
wavelet transform Wψ f .
Ω,k
Theorem 2.2. Let f ∈ LΩ,k
p and ψ ∈ Lp , 0 < p ≤ 1, then for the wavelet transform
Wψ f we have the estimates:
(2.7)
k (Wψ f )(a, b) kLkp ≤ Ca
α+ n
2
k f kLkp k ψ kLkp
for (2.2);
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
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(2.8)
(2.9)
k (Wψ f )(a, b) kLkp ≤ Ca
k (Wψ f )(a, b) k
k
Lpβ,γ
|α|+ n
2
n
2
≤ Ca e
k f kLkp k ψ kLkp
1
βa2γ
2
kf k
k
Lpβ,γ
for (2.3);
kψk
k
Lpβ,2γ
where b ∈ Rn and a > 0.
for (2.4),
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Proof. For k(x) = |x|α , α > 0,we have k(az) = aα k(z) and
Z
p1
x p
p
−n
kha,0 kLkp =
k (x)(a 2 |h( )|) dx
a
Rn
Z
p1
n
p
p
= a2
k (az)|h(z)| dz
Rn
=a
n
2
Z
Rn
apα k p (z)|h(z)|p dz
p1
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=a
n
+α
2
Z
p
p1
p
k (z)|h(z)| dz
Rn
n
= a 2 +α k h kLkp
n
= a 2 +α k ψ kLkp .
For k(x) =
Qn
j=1
|xj |αj , αj ≥ 0, we have k(az) = a|α| k(z) and
Wavelet Transform
R.S. Pathak and S.K. Singh
p1
x p
=
k (x)(a |h( )|) dx
a
Rn
Z
p1
n
p
p
k (az)|h(z)| dz
= a2
Z
kha,0 kLkp
−n
2
p
vol. 8, iss. 1, art. 9, 2007
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Rn
=a
n
2
Z
a
p|α| p
Contents
p1
p
k (z)|h(z)| dz
Rn
=a
=a
n
+|α|
2
=a
n
+|α|
2
β|x|γ
Next, for k(x) = kβ,γ (x) = e
γ
γ |z|γ
kβ,γ (az) = eβ|az| = eβa
n
+|α|
2
Z
p
p
p1
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k (z)|h(z)| dz
Rn
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k h kLkp
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k ψ kLkp .
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, β ≥ 0, 0 ≤ γ ≤ 1, we have
≤ eβ
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a2γ +|z|2γ
2
1
2γ
1
= e 2 βa e 2 β|z|
2γ
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1
2γ
= e 2 βa kβ,2γ (z),
and
Z
kha,0 k
k
Lpβ,γ
=
Rn
=a
n
2
Z
Rn
≤a
n
2
x p p1
a h
dz
a
p1
p
p
kβ,γ (az)|h(z)| dz
p
kβ,γ
(x)
Z
e
1
pβa2γ
2
Rn
n
1
2γ
= a 2 e 2 βa
1
βa2γ
2
n
2
1
βa2γ
2
=a e
=a e
p
kβ,2γ
(z)|h(z)|p dz
p1
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
Z
p
kβ,2γ
(z)|h(z)|p dz
Rn
n
2
−n
2
p1
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khk
k
Lpβ,2γ
kψk
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The proofs of (2.7), (2.8) and (2.9) follow from (2.6).
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k
Lpβ,2γ
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3.
Boundedness of W in Bp,k
The space Bp,k (Rn ) was introduced by Hörmander [1], as a generalization of the
Sobolev space H s (Rn ), in his study of the theory of partial differential equations.
We recall its definition.
Definition 3.1. A positive function k defined in Rn will be called a temperate weight
function if there exist positive constants C and N such that
(3.1)
N
k(ξ + η) ≤ (1 + C|ξ|) k(η);
n
ξ, η ∈ R ,
the set of all such functions k will be denoted by K . Certain properties of the weight
function k are contained in the following theorem whose proof can be found in [1].
Theorem 3.1. If k1 and k2 belong to K ,then k1 + k2 , k1 k2 , sup(k1 , k2 ), inf(k1 , k2 ),
are also in K . If k ∈ K we have k s ∈ K for every real s,and if µ is a positive
measure we have either µ ∗ k ≡ ∞ or else µ ∗ k ∈ K .
Definition 3.2. If k ∈ K and 1 ≤ p ≤ ∞, we denote by Bp,k the set of all distributions u ∈ S 0 such that û is a function and
p1
Z
−n
p
(3.2)
k u kp,k = (2π)
|k(ξ)û| dξ
< ∞, 1 ≤ p < ∞;
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
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(3.3)
k u k∞,k = ess sup |k(ξ)û(ξ)|.
We need the following theorem [1, p.10] in the proof of our boundedness result.
T
Theorem 3.2 (Lars Hörmander). If u1 ∈ Bp,k1 E 0 and u2 ∈ B∞,k2 then u1 ∗ u2 ∈
B∞,k1 k2 , and we have the estimate
(3.4)
k u1 ∗ u2 kp,k1 k2 ≤k u1 kp,k1 k u2 k∞,k2 ,
1 ≤ p < ∞.
Close
Using the above theorem we obtain the following boundedness result.
T
Theorem 3.3. Let k1 and k2 belong to K . Assume that f ∈ Bp,k1 E 0 and ψ ∈
B∞,k2 then the wavelet transform (Wψ f )(a, b) = (f ∗ ha,0 )(b), defined by (1.1) is in
Bp,k1 k2 , and
N
n
1
C 2
2
(3.5)
k Wψ f (a, b) kp,k1 k2 ≤ a k2
k f kp,k1 1 + t
ψ̂(t) .
2
2a
2
∞
Proof. Since
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
kha,0 k∞,k2 = ess sup k2 (ξ)ĥa,0 (ξ)
n
2
= ess sup k2 (ξ)a ψ̂(aξ)
n
≤ a 2 ess sup k2 ( at )ψ̂(t)
N
n
C
1
2
≤ a 2 k2
ess
sup
1
+
t
ψ̂(t)
2a2
2
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This proves the theorem.
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on using (3.1). Hence by Theorem 3.2 we have
kWψ f (a, b)kp,k1 k2 =k (f ∗ ha,0 (b) kp,k1 k2
n
1
k f kp,k1
≤ a 2 k2
2a2
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N
C
ψ̂(t) .
1 + t2
2
∞
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4.
A General Boundedness Result
Using Young’s inequality for convolution we obtained a general boundedness result
for the wavelet transform. In the proof of our result the following theorem will be
used [3, p. 90].
Theorem 4.1. Let p, q, r ≥ 1 and p1 + 1q + 1r = 2. Let k ∈ Lp (Rn ), f ∈ Lq (Rn ) and
g ∈ Lr (Rn ), then
Z
kf ∗ gk∞,k = k(x)(f ∗ g)(x)dx
n
ZR Z
= k(x)f (x − y)g(y)dxdy Rn
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
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Rn
Contents
≤ Cp,q,r;n k k kp k f kq k g kr .
1
n
The sharp constant Cp,q,r;n = (Cp Cq Cr ) , where
Cp2
=
pp
0 10
p
with
p
( p1
+
1
p0
= 1).
Using Theorem 4.1 and following the same method of proof as for Theorem 3.3 we
obtain the following boundedness result.
Theorem 4.2. Let p, q, r ≥ 1,
ψ ∈ Lr (Rn ), then
1
p
+ 1q +
1
r
= 2 and k ∈ Lp (Rn ). Let f ∈ Lq (Rn ) and
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n
n
k Wψ f k∞,k ≤ Cp,q,r;n a r − 2 k k kp k f kq k ψ kr
1
where Cp,q,r;n = (Cp Cq Cr )n , Cp2 =
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pp
1
p0 p0
with
1
p
+
1
p0
= 1.
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References
[1] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators II,
Springer-Verlag, Berlin Heiedelberg New York, Tokyo, 1983.
[2] T.H. KOORNWINDER, Wavelets, World Scientific Publishing Co. Pty. Ltd.,
Singapore , 1993.
[3] E.H. LIEB AND M. LOSS, Analysis, Narosa Publishing House, 1997. ISBN:
978-81-7319-201-2.
[4] R.S. PATHAK, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Science Publishers, Amsterdam, 1997.
Wavelet Transform
R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
Title Page
[5] R.S. PATHAK, The continuous wavelet transform of distributions, Tohoku
Math. J., 56 (2004), 411–421.
[6] H. TRIEBEL, A note on quasi-normed convolution algebras of entire analytic
functions of exponential type, J. Approximation Theory, 22(4) (1978), 368–373.
[7] H. TRIEBEL, Multipliers in Bessov-spaces and in LΩ
p - spaces (The cases 0 <
p ≤ 1 and p = ∞), Math. Nachr., 75 (1976), 229–245.
[8] H.J. SCHMEISSER AND H. TRIEBEL, Topics in Fourier Analysis and Function Spaces, John Wiley and Sons, Chichester, 1987.
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