Wavelet Transform R.S. Pathak and S.K. Singh
vol. 8, iss. 1, art. 9, 2007
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BOUNDEDNESS OF THE WAVELET TRANSFORM IN CERTAIN FUNCTION SPACES
R.S. PATHAK AND S.K. SINGH
Department of Mathematics, Banaras Hindu University, Varanasi - 221 005, India
EMail:ramshankarpathak@yahoo.co.in
Received: 06 October, 2005 Accepted: 25 January, 2007 Communicated by: L. Debnath 2000 AMS Sub. Class.: 42C40, 46F12.
Key words: Continuous wavelet transform, Distributions, Sobolev space, Besov space, Lizorkin-Triebel space.
Abstract: Using convolution transform theory boundedness results for the wavelet trans- form are obtained in the Triebel space-LΩ,kp , Hörmander space-Bp,q(Rn) and general function space-L∞,k,wherekdenotes a weight function possessing spe- cific properties in each case.
Wavelet Transform R.S. Pathak and S.K. Singh
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Contents
1 Introduction 3
2 Boundedness ofW inLΩ,kp 4
3 Boundedness ofW inBp,k 8
4 A General Boundedness Result 10
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1. Introduction
The wavelet transformW of a functionf with respect to the waveletψ is defined by
(1.1) f(a, b) = (W˜ ψf)(a, b) = Z
Rn
f(t)ψa,b(t)dt = (f ∗ha,0)(b),
whereψa,b =a−n2ψ(x−ba ), h(x) =ψ(−x),b ∈ Rnanda > 0, provided the integral exists. In view of (1.1) the wavelet transform (Wψf)(a, b) can be regarded as the convolution off andha,0. The existence of convolutionf ∗g has been investigated by many authors. For this purpose Triebel [6] defined the spaceLΩ,kp and showed that for certain weight functions k, f ∗g ∈ LΩ,kp , where f, g ∈ LΩ,kp , 0 < p ≤ 1. Convolution theory has also been developed by Hörmander in the generalized Sobolev spaceBp,q(Rn), 1≤p≤ ∞.
In Section2of the paper, a definition and properties of the spaceLΩ,kp are given and a boundedness result for the wavelet transformWψf is obtained. In Section3we recall the definition and properties of the generalized Sobolev space Bp,q(Rn) due to Hörmander [1] and obtain a certain boundedness result for Wψf. Finally, using Young’s inequality a third boundedness result is also obtained.
Wavelet Transform R.S. Pathak and S.K. Singh
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2. Boundedness of W in L
Ω,kpLet us recall the definition of the spaceLΩ,kp by Triebel [6].
Definition 2.1. Let Ω be a bounded C∞-domain in Rn. If k(x) is a non-negative weight function inRnand0< p≤ ∞, then
(2.1) LΩ,kp =
f|f ∈S0,suppF f ⊂Ω;
kf kLΩ,k
p =kkf kLp= Z
Rn
kp(x)|f(x)|pdx 1p
<∞
. Ifk(x) = 1thenLΩ,kp =LΩp.
We need the following theorem [6, p. 369] in the proof of our boundedness result.
Theorem 2.1 (Hans Triebel). Ifkis one of the following weight functions:
(2.2) k(x) =|x|α, α≥0
(2.3) k(x) =
n
Y
j=1
|xj|αj, αj ≥0
(2.4) k(x) = kβ,γ(x) =eβ|x|γ, β ≥0,0≤γ ≤1 and0< p≤1, then
(2.5) LΩ,kp ∗LΩ,kp ⊂LΩ,kp
Wavelet Transform R.S. Pathak and S.K. Singh
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and there exists a positive numberCsuch that for allf, g ∈LΩ,kp ,
(2.6) kf ∗gkLk
p ≤CkfkLk pkgkLk
p.
Using the above theorem we obtain the following boundedness result for the wavelet transformWψf.
Theorem 2.2. Letf ∈LΩ,kp andψ ∈LΩ,kp ,0< p ≤1,then for the wavelet transform Wψf we have the estimates:
(2.7) k(Wψf)(a, b)kLk
p≤Caα+n2 kf kLk
pkψ kLk
p for (2.2);
(2.8) k(Wψf)(a, b)kLk
p≤Ca|α|+n2 kf kLk
pkψ kLk
p for (2.3);
(2.9) k(Wψf)(a, b)k
Lkβ,γp
≤Can2e12βa2γ kf k
Lkβ,γp
kψ k
Lkβ,2γp
for (2.4), whereb ∈Rnanda >0.
Proof. Fork(x) = |x|α, α >0,we havek(az) = aαk(z)and kha,0kLk
p = Z
Rn
kp(x)(a−n2|h(x a)|)pdx
1p
=an2 Z
Rn
kp(az)|h(z)|pdz 1p
=an2 Z
Rn
apαkp(z)|h(z)|pdz 1p
Wavelet Transform R.S. Pathak and S.K. Singh
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=an2+α Z
Rn
kp(z)|h(z)|pdz 1p
=an2+α khkLkp
=an2+α kψ kLk
p . Fork(x) = Qn
j=1|xj|αj, αj ≥0, we havek(az) = a|α|k(z)and kha,0kLk
p = Z
Rn
kp(x)(a−n2|h(x a)|)pdx
1p
=an2 Z
Rn
kp(az)|h(z)|pdz 1p
=an2 Z
Rn
ap|α|kp(z)|h(z)|pdz 1p
=an2+|α|
Z
Rn
kp(z)|h(z)|pdz 1p
=an2+|α|khkLk
p
=an2+|α|kψ kLkp .
Next, fork(x) = kβ,γ(x) =eβ|x|γ, β ≥0,0≤γ ≤1, we have kβ,γ(az) = eβ|az|γ =eβaγ|z|γ ≤eβa
2γ+|z|2γ
2 =e12βa2γe12β|z|2γ =e12βa2γkβ,2γ(z),
Wavelet Transform R.S. Pathak and S.K. Singh
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and
kha,0k
Lkβ,γp
= Z
Rn
kβ,γp (x) a−n2
hx
a
p
dz 1p
=an2 Z
Rn
kpβ,γ(az)|h(z)|pdz 1p
≤an2 Z
Rn
e12pβa2γkβ,2γp (z)|h(z)|pdz 1p
=an2e12βa2γ Z
Rn
kpβ,2γ(z)|h(z)|pdz 1p
=an2e12βa2γ kh k
Lkβ,2γp
=an2e12βa2γ kψ k
Lkβ,2γp
. The proofs of (2.7), (2.8) and (2.9) follow from (2.6).
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3. Boundedness of W in B
p,kThe space Bp,k(Rn) was introduced by Hörmander [1], as a generalization of the Sobolev space Hs(Rn), in his study of the theory of partial differential equations.
We recall its definition.
Definition 3.1. A positive functionkdefined inRnwill be called a temperate weight function if there exist positive constantsCandN such that
(3.1) k(ξ+η)≤(1 +C|ξ|)Nk(η); ξ, η∈Rn,
the set of all such functionskwill be denoted byK . Certain properties of the weight functionkare contained in the following theorem whose proof can be found in [1].
Theorem 3.1. Ifk1andk2belong toK ,thenk1+k2, k1k2,sup(k1, k2),inf(k1, k2), are also inK . If k ∈ K we have ks ∈ K for every reals,and ifµis a positive measure we have eitherµ∗k ≡ ∞or elseµ∗k ∈K .
Definition 3.2. Ifk ∈ K and1≤p≤ ∞, we denote byBp,kthe set of all distribu- tionsu∈S0 such thatuˆis a function and
(3.2) kukp,k= (2π)−n Z
|k(ξ)ˆu|pdξ 1p
<∞, 1≤p < ∞;
(3.3) kuk∞,k=esssup|k(ξ)ˆu(ξ)|.
We need the following theorem [1, p.10] in the proof of our boundedness result.
Theorem 3.2 (Lars Hörmander). Ifu1 ∈Bp,k1TE0andu2 ∈B∞,k2 thenu1∗u2 ∈ B∞,k1k2, and we have the estimate
(3.4) ku1∗u2 kp,k1k2≤ku1 kp,k1ku2 k∞,k2, 1≤p < ∞.
Wavelet Transform R.S. Pathak and S.K. Singh
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Using the above theorem we obtain the following boundedness result.
Theorem 3.3. Let k1 and k2 belong to K . Assume that f ∈ Bp,k1TE0 andψ ∈ B∞,k2 then the wavelet transform(Wψf)(a, b) = (f∗ha,0)(b),defined by (1.1) is in Bp,k1k2, and
(3.5) kWψf(a, b)kp,k1k2≤an2k2 1
2a2
kf kp,k1
1 + C
2t2 N
ψˆ(t) ∞
.
Proof. Since
kha,0k∞,k
2 =esssup
k2(ξ)ˆha,0(ξ)
=esssup
k2(ξ)an2ψ(aξ)ˆ
≤an2esssup
k2(at) ˆψ(t)
≤an2k2 1
2a2
esssup
1 + C
2t2 N
ψ(t)ˆ
on using (3.1). Hence by Theorem3.2we have kWψf(a, b)kp,k
1k2 =k(f∗ha,0(b)kp,k1k2
≤an2k2 1
2a2
kf kp,k1
1 + C
2t2 N
ψ(t)ˆ ∞
. This proves the theorem.
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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. Letp, q, r ≥1and 1p +1q +1r = 2. Letk ∈Lp(Rn), f ∈Lq(Rn)and g ∈Lr(Rn), then
kf ∗gk∞,k = Z
Rn
k(x)(f ∗g)(x)dx
= Z
Rn
Z
Rn
k(x)f(x−y)g(y)dxdy
≤Cp,q,r;n kkkpkf kqkg kr . The sharp constant Cp,q,r;n = (CpCqCr)n, where Cp2 = p
1 p
p0
1
p0 with (1p + p10 = 1).
Using Theorem4.1and following the same method of proof as for Theorem3.3we obtain the following boundedness result.
Theorem 4.2. Letp, q, r ≥1, 1p + 1q +1r = 2andk∈Lp(Rn). Letf ∈Lq(Rn)and ψ ∈Lr(Rn), then
kWψf k∞,k ≤ Cp,q,r;nanr−n2 kk kpkf kqkψ kr whereCp,q,r;n = (CpCqCr)n,Cp2 = p
p1
p0
1
p0 with 1p + p10 = 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 Appli- cations, Gordon and Breach Science Publishers, Amsterdam, 1997.
[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 inLΩp- spaces (The cases0 <
p≤1andp=∞), Math. Nachr., 75 (1976), 229–245.
[8] H.J. SCHMEISSERANDH. TRIEBEL, Topics in Fourier Analysis and Func- tion Spaces, John Wiley and Sons, Chichester, 1987.