ON THE CONTINUOUS WAVELET TRANSFORM ... - Semantic Scholar
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International Journal of Wavelets, Multiresolution and Information Processing Vol. 10, No. 4 (2012) 1250038 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0219691312500385
ON THE CONTINUOUS WAVELET TRANSFORM ON HOMOGENEOUS SPACES
F. ESMAEELZADEH∗,‡ , R. A. KAMYABI GOL†,§ and R. RAISI TOUSI∗,¶ ∗Department
of Pure Mathematics, Ferdowsi University of Mashhad P. O. Box 1159-91775, Mashhad, Iran
†Department
of Pure Mathematics, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS) P. O. Box 1159-91775, Mashhad, Iran ‡[email protected] §[email protected] ¶[email protected] Received 6 April 2011 Revised 8 November 2011 Published 14 July 2012
Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space H. The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto. Keywords: Homogeneous space; square integrable representation; admissible wavelet; continuous wavelet transform; reproducing kernel Hilbert space. AMS Subject Classification: Primary 43A15; Secondary 43A85, 65T60
1. Introduction and Preliminaries The wavelet transform has been an alternative to time frequency analysis. Like time frequency, wavelet transform analysis is a prominent tool in mathematics, physics, applied sciences, signal and data processing.19,20 The continuous wavelet transform on Rd is studied by several authors such as Refs. 4, 5, 7, 8 and 17 and generalized to locally compact groups and semidirect product groups by Refs. 1, 3, 9, 12–14. Ali, Antoine and Gazeau in Ref. 1 have studied continuous wavelet transform on a homogeneous space G/H using Borel sections, where G is a locally compact group and H is a closed subgroup of G, assuming a G-invariant measure on G/H (see Ref. 1, Sec. 9.2). In this paper we study continuous wavelet transform on a 1250038-1
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homogeneous space G/H where G is a locally compact group and H is a compact subgroup of G. We do this task with a completely different approach considering a relatively invariant measure on G/H. Let G be a locally compact group and H be a closed subgroup of G. Consider G/H as a homogeneous space on which G acts from the left and µ as a Radon measure on it (for a detailed account of homogeneous spaces, the reader is referred to Refs. 10, 11 and 18. For g ∈ G and Borel subset E of G/H, we define the translation µg of µ by µg (E) = µ(gE). A measure µ is said to be G-invariant if µg = µ, for all g ∈ G. The measure µ is said to be strongly quasi-invariant provided that a continuous function λ : G × G/H → (0, ∞) exists which satisfies dµg (kH) = λ(g, kH)dµ(kH), for all g, k ∈ G. If the function λ(g, ·) reduces to constant, then µ is called relatively invariant under G. A rho-function for the pair (G, H) is defined to be a continuous function ρ : G → (0, ∞) which satisfies ρ(gh) =
∆H (h) ρ(g) (g ∈ G, h ∈ H), ∆G (h)
where ∆G , ∆H are the modular functions on G and H, respectively. It is well known11 : any pair (G, H) admits a rho-function and for each rho-function ρ there is a strongly quasi-invariant measure µ on G/H such that ρ(gk) dµg (kH) = dµ ρ(k)
(g, k ∈ G).
Moreover, the existence of a homomorphism rho-function for the pair (G, H) is a necessary and sufficient condition for the existence of a relatively invariant measure on G/H. Let µ be a relatively invariant measure on G/H which arises from a rho-function ρ. It has been shown in Ref. 15 ρ(gk) =
ρ(g)ρ(k) ρ(e)
and ρ(g) dµg (kH) = , dµ ρ(e) for all g, k ∈ G. If H is a compact subgroup of G then ∆G is a homomorphism rho-function and hence G/H admits a relatively invariant measure µ which arises from a ρ-function. In the sequel we shall consider the space Lp (G/H) (1 ≤ p ≤ ∞) with respect to this measure. In this paper we need to define the basic concepts in the theory of unitary representation of homogeneous spaces. A continuous unitary representation of the homogeneous space G/H is a map σ from G/H into the group U (H), of all unitary operators on some nonzero Hilbert space H, for which gH → σ(gH)x, y is 1250038-2
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continuous from G/H into C, for each x, y ∈ H and σ(gkH) = σ(gH)σ(kH),
σ(g −1H) = σ(gH)∗ ,
for each g, k ∈ G. This defines a continuous unitary representation π of G in which the subgroup H is considered to be contained in kernel of π. Conversely, any continuous unitary representation π of G which is trivial on H induces a continuous unitary representation σ of G/H, by letting σ(gH) = π(g). Moreover, a closed subspace M of H is said to be invariant with respect to σ if σ(gH)M ⊆ M , for all g ∈ G. A continuous unitary representation σ is said to be irreducible if the only invariant subspaces of H are {0} and H. (In the sequel we always mean by a representation, a continuous unitary representation.) Let G be a locally compact group and H be a compact subgroup of G. Then as mentioned above, G/H possesses a relatively invariant measure µ. The basic example of a representation of G which arises from the natural left action of G on G/H and we call it the left type regular representation, is defined as 1/2 ρ(e) Lg ψ, (1.1) : G → U (L2 (G/H)), (g)ψ = ρ(g) where Lg ψ(kH) = ψ(g −1 kH) and L2 (G/H) is L2 (G/H, µ), where µ is a relatively invariant measure on G/H which arises from a rho-function ρ. In this paper we study the continuous wavelet transform on G/H and give a reconstruction formula for it. Among other things we show that the continuous wavelet transform is onto if and only if L2 (G/H) is a reproducing kernel Hilbert space. Our construction is useful since it is a general approach to the continuous wavelet transform on homogeneous spaces. It includes most of examples on locally compact groups, semidirect product groups as special cases and an example of a homogeneous space which is not a semidirect product of groups (see examples in Sec. 5). This paper is organized as follows: In Sec. 2 we define a square integrable representation of G/H and prove the reconstruction formula for it. As a result, we show that the set of admissible vectors is path connected (even the set of admissible vectors is a subspace). Section 3 is devoted to introducing the continuous wavelet transform on G/H. We prove that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we show that the continuous wavelet transform is onto if and only if L2 (G/H) is a reproducing kernel Hilbert space. Some properties of continuous wavelet transform on G/H is discussed in Sec. 4. It is shown that the continuous wavelet transform is a filter and so a multiplier. Finally, in Sec. 5 we present some examples which reveal the application of our results. 2. Square Integrable Representation of G/H Throughout this paper we assume that G is a locally compact group and H is a compact subgroup of G. Let H be a Hilbert space and σ be an irreducible representation 1250038-3
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of G/H on H. We aim to define a square integrable representation of G/H, an admissible vector and an admissible wavelet for such a representation. Definition 2.1. Let G be a locally compact group and H be a compact subgroup of G. An irreducible representation σ of G/H on H is said to be square integrable if there exists a nonzero element ζ ∈ H such that ρ(e) |ζ, σ(gH)ζ|2 dµ(gH) < ∞, (2.1) ρ(g) G/H where µ is a relatively invariant measure on G/H which arises from a rho-function ρ : G → (0, ∞). If ζ satisfies (2.1), it is called an admissible vector. An admissible vector ζ ∈ H is called an admissible wavelet if ζ = 1. In this case, we define the wavelet constant cζ as cζ := G/H
ρ(e) |ζ, σ(gH)ζ|2 dµ(gH). ρ(g)
(2.2)
We call cζ the wavelet constant associated to the admissible wavelet ζ. Remark 2.1. Note that since H is a compact subgroup of G, ∆G |H = ∆H = 1. So, for every rho-function ρ, we have ρ(gh) = ρ(g) for all g ∈ G and h ∈ H. This implies that there is a function ρ˜ on G/H such that for each g ∈ G, ρ˜(gH) = ρ(g). Therefore Definition 2.1 is well defined. We now prove a reconstruction formula for square integrable representation σ of G/H. To this end, we need an auxiliary lemma (Lemma 2.3) in which, we use a straightforward generalization of the classical Schur’s lemma (see Ref. 1, Lemma 4.3.1) as follows: Lemma 2.1 (Classical Schur’s lemma for homogeneous spaces). Let G be a locally compact group and H be a closed subgroup of G. Let H be a Hilbert space and σ be an irreducible representation of G/H on H. If T is a bounded operator on H which commutes with σ(gH), for all g ∈ G, i.e. T σ(gH) = σ(gH)T, then T = λI, for λ ∈ C. Similarly, it is easily shown that the extended Schur’s lemma (see Ref. 1, Lemma 4.3.3) is generalized to the homogeneous space setting as follows: Lemma 2.2 (Extended Schur’s lemma for homogeneous spaces). Let σ1 be an irreducible representation of G/H on a Hilbert space H1 and σ2 be a representation of G/H on a Hilbert space H2 . Let T : H1 → H2 be a densely defined closed linear map and the domain T is invariant under σ1 , and suppose that T intertwines σ1 and σ2 , i.e. T σ1 (gH) = σ2 (gH)T . Then T is either null or a multiple of an isometry. 1250038-4
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Lemma 2.3. Let σ be a square integrable representation of G/H, and ζ be an admissible vector for σ. Then ρ(e) |x, σ(gH)ζ|2 dµ(gH) < ∞ (2.3) M = x ∈ H, G/H ρ(g) is a closed subspace of H. Proof. Let ζ be an admissible vector and M be defined as in (2.3). Then M is invariant under σ. Indeed for all x ∈ M and k ∈ G we have ρ(e) |σ(kH)x, σ(gH)ζ|2 dµ(gH) ρ(g) G/H ρ(e) |x, σ(k −1 gH)ζ|2 dµ(gH) = ρ(g) G/H ρ(e) |x, σ(gH)ζ|2 dµk (gH) = ρ(kg) G/H ρ(e) |x, σ(gH)ζ|2 dµ(gH) < ∞. = ρ(g) G/H Since M is invariant so is M . By the irreducibility of σ we have M = H. Define the linear operator 1/2 ρ(e) x, σ(gH)ζ. (2.4) Aζ : M → L2 (G/H), Aζ (x)(gH) = ρ(g) We prove that Aζ is a closed operator. To this end, take a sequence {xn }n∈N ⊆ M, converging to x ∈ H such that Aζ xn converges in L2 (G/H) to an element ϕ ∈ L2 (G/H). Then there exists a subsequence of Aζ xn , again denoted by Aζ xn , which converges to ϕ a.e. on G/H. By the continuity of the inner product in H we have 1/2 1/2 ρ(e) ρ(e) xn , σ(gH)ζ → x, σ(gH)ζ, ρ(g) ρ(g) 1/2 ρ(e) x, σ(gH)ζ = ϕ(gH). Moreover, for all g ∈ G. Thus ρ(g) Aζ σ(gH) = (g)Aζ , where is a left type regular representation as defined in (1.1). Indeed, 1/2 ρ(e) (Aζ σ(gH)x)(kH) = σ(gH)x, σ(kH)ζ ρ(k) 1/2 ρ(e) = x, σ(g −1kH)ζ ρ(k) 1250038-5
(2.5)
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= = =
ρ(e) ρ(g) ρ(e) ρ(g) ρ(e) ρ(g)
1/2 1/2
ρ(e) ρ(g −1 k)
1/2
x, σ(g −1kH)ζ
(Aζ x)(g −1kH)
1/2 Lg (Aζ x)(kH)
= (g)(Aζ x)(kH). Now M equipped with the norm · ζ , defined as x 2ζ = x 2 + Aζ x 2L2 (G/H) , is a Hilbert space. So extended Schur’s lemma implies that Aζ from M into L2 (G/H) is a multiple of an isometry. That is there exists a positive constant λ such that, Aζ x 2L2 (G/H) = λ x 2ζ = λ x 2 + λ Aζ x 2L2 (G/H) . Hence Aζ x 2L2 (G/H) = λ 2 1−λ x , for all x ∈ M and λ < 1. Since M is dense in H we can extend ζ from H into L2 (G/H). Aζ : M → L2 (G/H) to a bounded linear operator A Let {xn }n∈N ⊆ M such that xn → x in H as n → ∞. Then Aζ xn → A˜ζ x in
L2 (G/H) as n → ∞. Since Aζ is a closed linear operator from H into L2 (G/H) with domain M , it follows that x ∈ M , i.e. M is a closed subspace of H.
Now we can state our main theorem in this section, which is the reconstruction formula. Theorem 2.1. Let σ be a square integrable representation of G/H on H. If ζ is an admissible wavelet for σ, then ρ(e) 1 x, σ(gH)ζσ(gH)ζ, ydµ(gH), (2.6) x, y = cζ G/H ρ(g) where cζ is as in (2.2). Proof. Let M be as in (2.3). By Lemma 2.3 M is a closed subspace of H. As has been shown in the proof of the lemma, M is invariant under σ and by irreducibility of σ, it follows that M = H. So Aζ is a bounded linear operator on H. Now A∗ζ Aζ intertwines with σ i.e. σ(gH)A∗ζ Aζ = A∗ζ Aζ σ(gH). Indeed, A∗ζ Aζ σ(gH)x, y = A∗ζ (g)Aζ x, y = (g)Aζ x, Aζ y = Aζ x, (g −1 )Aζ y = Aζ x, Aζ σ(g −1 H)y = σ(gH)A∗ζ Aζ x, y, where we have used (2.5) in the above equalities. By Lemma 2.1 there exists a constant c such that A∗ζ Aζ = cI, where I is the identity operator on H. Therefore 1250038-6
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we get cx, y = A∗ζ Aζ x, y = Aζ x, Aζ y Aζ x(gH)Aζ y(gH)dµ(gH) = G/H
=
G/H
ρ(e) x, σ(gH)ζσ(gH)ζ, ydµ(gH), ρ(g)
which is the desired identity. If we put x = y = ζ then ρ(e) c= |ζ, σ(gH)ζ|2 dµ(gH) = cζ . G/H ρ(g) As an interesting consequence of Theorem 2.1 we conclude that the set of admissible vectors is path connected. Corollary 2.1. The set of admissible vectors is path connected. Proof. Let ζ and ξ be two admissible vectors in H. By the following calculations, we get ζ + ξ is an admissible vector: ρ(e) |ζ + ξ, σ(gH)(ζ + ξ)|2 dµ(gH) ρ(g) G/H ρ(e) |ζ, σ(gH)ζ|2 dµ(gH) ≤ ρ(g) G/H ρ(e) |ζ, σ(gH)ξ|2 dµ(gH) + ρ(g) G/H ρ(e) |ξ, σ(gH)ζ|2 dµ(gH) + ρ(g) G/H ρ(e) |ξ, σ(gH)ξ|2 dµ(gH) + ρ(g) G/H ρ(e) |ζ, σ(gH)ζξ, σ(gH)ζ|dµ(gH) +2 ρ(g) G/H ρ(e) + |ζ, σ(gH)ζζ, σ(gH)ξ|dµ(gH) G/H ρ(g) ρ(e) |ζ, σ(gH)ζξ, σ(gH)ξ|dµ(gH) + G/H ρ(g) ρ(e) + |ξ, σ(gH)ζζ, σ(gH)ξ|dµ(gH) G/H ρ(g) 1250038-7
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+ G/H
+ G/H
ρ(e) |ξ, σ(gH)ζξ, σ(gH)ξ|dµ(gH) ρ(g)
ρ(e) |ζ, σ(gH)ξξ, σ(gH)ξ|dµ(gH) . ρ(g)
Since ξ is an admissible vector, Theorem 2.1 implies that ρ(e) |ζ, σ(gH)ξ|2 dµ(gH) ρ(g) G/H ρ(e) ζ, σ(gH)ξσ(gH)ξ, ζdµ(gH) = ρ(g) G/H = cζ ζ, ζ = cζ ζ 2 < ∞. Similarly, from the admissibility of ζ, we have ρ(e) |ξ, σ(gH)ζ|2 dµ(gH) < ∞. ρ(g) G/H Also by Schawrz inequality we get ρ(e) |ζ, σ(gH)ζξ, σ(gH)ζ|dµ(gH) G/H ρ(g) 1/2 ρ(e) 2 |ζ, σ(gH)ζ| dµ(gH) ≤ G/H ρ(g) × G/H
1/2 ρ(e) 2 |ξ, σ(gH)ζ| dµ(gH) < ∞. ρ(g)
So G/H
ρ(e) |ζ + ξ, σ(gH)(ζ + ξ)|2 dµ(gH) < ∞. ρ(g)
Therefor ζ + ξ is an admissible vector. In particular αζ + (1 − α)ξ is an admissible vector, which completes the proof. As an immediate consequence of Corollary 2.1, the set of admissible vectors is a linear space. In fact, we have the following corollary. Corollary 2.2. With notations as above, the set of all admissible vectors is a subspace of H. 1250038-8
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3. The Continuous Wavelet Transform on G/H In this section our notation will be the same as before. We define the continuous wavelet transform on the homogeneous space G/H and show that the range of continuous wavelet transform on G/H is a reproducing kernel Hilbert space. Also, it is shown that continuous wavelet transform is onto if L2 (G/H) is a reproducing kernel Hilbert space. Definition 3.1. Let σ be a representation of G/H on a Hilbert space H and ζ be an admissible wavelet for σ. We define the continuous wavelet transform associated to the admissible wavelet ζ as the linear operator Wζ : H → C(G/H) defined by 1/2 ρ(e) 1 x, σ(gH)ζ, (Wζ x)(gH) = √ cζ ρ(g) for all x ∈ H, g ∈ G where cζ is the wavelet constant associated to ζ as in (2.2). Also ζ ∈ H is called bounded if Wζ is bounded. Note that if σ is a square integrable representation of G/H on H and ζ is an admissible wavelet for σ, then Wζ is a bounded linear operator from H into L2 (G/H). As an application of Theorem 2.1 we can easily conclude that Wζ is an isometry. Proposition 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then (Wζ x)(gH)(Wζ y)(gH)dµ(gH), (3.1) x, y = G/H
for every x, y ∈ H. In particular, the wavelet transform Wζ : H → L2 (G/H) is an isometry. Proof. By Theorem 2.1 the proof is obvious. As a consequence of Proposition 3.1 we conclude that the range of the continuous wavelet transform is a Hilbert space. Proposition 3.2. Let σ be a square integrable representation of G/H on H. Then the range of continuous wavelet transform Wζ : H → L2 (G/H) is a Hilbert space. Proof. Denote by Rang(Wζ ) the range of Wζ . Let {ϕj }∞ j=1 ⊆ Rang(Wζ ) be such ⊆ H such that Wζ xj = ϕj , that ϕj → ϕ in L2 (G/H) as j → ∞. Choose {xj }∞ j=1 for j = 1, 2, . . . . By Proposition 3.1 we have xj − xk = Wζ xj − Wζ xk L2 (G/H) = ϕj − ϕk L2 (G/H) → 0,
j, k = 1, 2, . . . .
Therefore {xj }∞ j=1 is a Cauchy sequence in H. So xj → x for some x ∈ H as j → ∞. Since Wζ is bounded, Wζ xj → Wζ x in L2 (G/H). Thus ϕ = Wζ x, i.e. ϕ ∈ Rang(Wζ ) and the proof is complete. 1250038-9
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At this point we recall that a reproducing kernel on a measure space (B, µ) is a function k from B × B to C such that (i) k(a, a) > 0, (ii) k(a, b) = k(b, a), (iii) k(a, c) = k(a, τ )k(τ, c)dµ(τ ),
(3.2) (3.3) (3.4)
B
for all a, b, c ∈ B. A Hilbert space H of functions on a measure space B is called a reproducing kernel Hilbert space if there exists a reproducing kernel k such that x(b)k(a, b)dµ(b), (3.5) x(a) = B
for all x ∈ H, a ∈ B. We are going to show that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. In fact, we show first in the following lemma that the function pζ : G/H × G/H → C defined by pζ (gH, kH) :=
ρ(e) pζ (g −1 kH) ρ(g)
is a reproducing kernel on G/H, where 1/2 1 ρ(e) pζ (gH) = σ(gH)ζ, ζ, cζ ρ(g)
(3.6)
(3.7)
for all ζ ∈ H, g ∈ G. In Theorem 3.1 we will prove that condition (3.5) is satisfied for Rang(Wζ ). To prepare for that we need the following lemma. Lemma 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then pζ defined as in (3.6) is a reproducing kernel on the homogeneous space G/H. Proof. Condition (3.2) obviously holds, i.e. p˜ζ (gH, gH) > 0. For (3.3), the following calculations show that p˜ζ (gH, kH) = p˜ζ (kH, gH). Indeed, 1/2 ρ(e) 1 ρ(e) · p˜ζ (gH, kH) = · σ(kH)ζ, σ(gH)ζ cζ ρ(g) ρ(g−1k) 1 1/2 1 ρ(e) 2 ρ(e) σ(kH)ζ, σ(gH)ζ = cζ ρ(g) ρ(k) 1/2 ρ(e) 1 ρ(e) = σ(gH)ζ, σ(kH)ζ cζ ρ(k) ρ(k −1 g) = p˜ζ (kH, gH). 1250038-10
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Moreover, by Theorem 2.1, pζ satisfies (3.4). Indeed, p˜ζ (gH, tH)˜ pζ (tH, kH)dµ(tH) G/H
ρ(e) ρ(e) pζ (g −1tH) pζ (t−1kH)dµ(tH) ρ(g) ρ(t) G/H 1/2 ρ(e) ρ(e) 1 = 2 cζ G/H ρ(gt) ρ(g −1 t) 1/2 ρ(e) × σ(tH)ζ, σ(gH)ζσ(kH)ζ, σ(tH)ζdµ(tH) ρ(t−1 k) 1/2 1 ρ(e) ρ(e) = cζ ρ(g) ρ(g −1 k) 1 ρ(e) σ(tH)ζ, σ(gH)ζσ(kH)ζ, σ(tH)ζdµ(tH) × c G/H ζ ρ(t) 1/2 1 ρ(e) ρ(e) = σ(kH)ζ, σ(gH)ζ cζ ρ(g) ρ(g −1 k) =
=
ρ(e) pζ (g −1kH) ρ(g)
= p˜ζ (gH, kH). Theorem 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then Rang(Wζ ) is a reproducing kernel Hilbert space with reproducing kernel p˜ζ as in (3.6) such that, ψ(gH) = ψ(kH)˜ pζ (gH, kH)dµ(kH). (3.8) G/H
Proof. By Lemma 3.1 it is enough to show (3.8). Let ψ ∈ Rang(Wζ ). Then there exists an element x in H such that ψ = Wζ x. Using Theorem 2.1 we get, ψ(kH)˜ pζ (gH, kH)dµ(kH) G/H
=
ρ(e) ρ(g)
ψ(kH)pζ (g −1 kH)dµ(kH) G/H
ρ(e) ψ(gkH)pζ (kH)dµg (kH) ρ(g) G/H 1/2 ρ(e) 1 = (Wζ x)(gkH)σ(kH)ζ, ζdµ(kH) cζ G/H ρ(k)
=
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=
=
1 3/2 cζ
1 1/2 cζ
G/H
ρ(e) ρ(g)
ρ(e) ρ(g)
1/2
1/2
ρ(e) x, σ(gkH)ζσ(kH)ζ, ζdµ(kH) ρ(k)
σ(g −1 H)x, ζ
= (Wζ x)(gH) = ψ(gH). In the sequel we intend to establish a condition under which Wζ is onto. Theorem 3.2. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. If ψ ∈ L2 (G/H) and ρ(e) ψ(kH)pζ (g −1kH)dµ(kH), ψ(gH) = ρ(g) G/H where pζ for g ∈ G is as defined in (3.7) then ψ ∈ Rang(Wζ ). Proof. For every ψ ∈ L2 (G/H), the expression 1/2 1 ρ(e) ψ(gH)σ(gH)ζdµ(gH) √ cζ G/H ρ(g) defines a vector in H and the integral is weakly convergent. To prove it, note that for any x ∈ H the function 1/2 ρ(e) gH → x, σ(gH)ζ ρ(g) is in L2 (G/H) and since ψ ∈ L2 (G/H), we have the integral
1/2
1 ρ(e)
ψ(gH)x, σ(gH)ζdµ(gH)
√
cζ G/H ρ(g)
= Wζ x(gH)ψ(gH)dµ(gH)
G/H
≤ Wζ x L2 (G/H) ψ L2 (G/H) . Then by the Riesz theorem,13 the integral 1/2 1 ρ(e) ψ(gH)σ(gH)ζdµ(gH) √ cζ G/H ρ(g) ρ(e) defines a vector in H. Now, if ψ(gH) = ρ(g) ψ(kH)pζ (g −1 kH)dµ(kH), then G/H 1/2 1 ρ(e) ψ(gH)σ(gH)ζdµ(gH) (kH) = ψ(kH). (3.9) Wζ √ cζ G/H ρ(g) 1250038-12
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Indeed, the following calculations complete the proof: 1/2 1 ρ(e) Wζ √ ψ(gH)σ(gH)ζdµ(gH) (kH) cζ G/H ρ(g) =
1 cζ
G/H
ρ(e) ρ(g)
1/2
ρ(e) ρ(k)
1/2 ψ(gH)σ(gH)ζ, σ(kH)ζdµ(gH)
1/2 ρ(e) ψ(gH)σ(k −1gH)ζ, ζdµ(gH) ρ(gk) G/H 1/2 ρ(e) 1 ρ(e) = ψ(gH)σ(k −1gH)ζ, ζdµ(gH) cζ ρ(k) G/H ρ(k −1 g) ρ(e) = ψ(gH)pζ (k −1gH)dµ(gH) ρ(k) G/H
=
1 cζ
= ψ(kH). Corollary 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. The continuous wavelet transform Wζ is onto if and only if L2 (G/H) is reproducing kernel Hilbert space. Proof. By Theorems 3.2, 3.1 the proof is obvious. Corollary 3.2. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then 1/2 ρ(e) 1 ∗ ψ(gH)σ(gH)ζdµ(gH), (3.10) Wζ (ψ) = √ cζ G/H ρ(g) for all ψ ∈ Dom(Wζ−1 ). Proof. Noticing Wζ is a unitary operator on its range, we have Wζ∗ = Wζ−1 . Now (3.9) implies the equality (3.10). 4. Some Properties of the Continuous Wavelet Transform on G/H As before let G be a locally compact group and H be a compact subgroup of G. If ζ is an admissible wavelet for the left type regular representation of G defined as in (1.1), then the continuous wavelet transform is a filter on L2 (G/H). That is, it intertwines with the left type regular representation. Proposition 4.1. Let be the left type regular representation of G on L2 (G/H) and ζ be an admissible wavelet for . Then the continuous wavelet transform Wζ : L2 (G/H) → L2 (G/H) is a filter. 1250038-13
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Proof. By the definition of continuous wavelet transform and left type regular representation of G/H we get, 1/2 ρ(e) 1 (Wζ (k)ψ)(gH) = √ (k)ψ, (g)ζ cζ ρ(g) 1/2 ρ(e) 1 = √ ψ, (k −1 g)ζ cζ ρ(g) 1/2 1/2 ρ(e) ρ(e) 1 = √ ψ, (k −1 g)ζ cζ ρ(k) ρ(k −1 g) 1/2 ρ(e) = Wζ ψ(k −1gH) ρ(k) 1/2 ρ(e) = Lk Wζ ψ(gH) ρ(k) = ((k)Wζ ψ)(gH), for all ψ ∈ L2 (G/H), k, g ∈ G. Since the continuous wavelet transform is a filter, now we intend to show that the continuous wavelet transform is a multiplier. First we need to mention Lp (G/H) is a Banach left L1 (G)-module under following action: For all 1 ≤ p ≤ ∞, there exists a left module action of L1 (G) on Lp (G/H) defined by f (k) 1/p f ∗p ψ(gH) = ρ(e) ψ(k −1gH)dk, 1/p G ρ(k) where f ∈ L1 (G) and ψ ∈ Lp (G/H). This action is well defined and under which Lp (G/H), 1 ≤ p ≤ ∞, is a Banach left L1 (G)-module with an approximate identity.15 Proposition 4.2. Let be the left type regular representation of G on L2 (G/H) and ζ be an admissible wavelet for . Then for all f ∈ L1 (G), ψ ∈ L2 (G/H) Wζ (f ∗2 ψ) = f ∗2 Wζ (ψ). Proof. Using Fubini’s theorem we get Wζ (f ∗2 ψ)(gH) 1/2 ρ(e) 1 f ∗2 ψ, (g)ζ =√ cζ ρ(g) 1/2 ρ(e) 1 (f ∗2 ψ)(kH)(g)ζ(kH)dµ(kH) =√ cζ ρ(g) G/H 1250038-14
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1 =√ cζ
ρ(e) ρ(g)
1/2 G/H
G
ρ(e) ρ(g)
1/2
ρ(e) ρ(t)
1/2
× f (t)ψ(t−1kH)ζ(g −1 kH)dtdµ(kH) 1 =√ cζ 1 =√ cζ
G
G
G/H
=
G
ρ(e) ρ(t)
ρ(e) ρ(g)
1/2
ρ(e) ρ(g)
1/2
1/2
ρ(e) ρ(t)
ρ(e) ρ(t)
1/2 f (t)Lt ψ(kH)(g)ζ(kH)dµ(kH)dt
1/2 f (t)Lt ψ, (g)ζdt
f (t)Wζ ψ(t−1gH)dt
= f ∗2 Wζ (ψ)(gH), for each f ∈ L1 (G), ψ ∈ L2 (G/H). Next, we show that admissibility and boundedness of an element ζ ∈ H is invariant under orthogonal projections. For showing this first we need the following lemma: Lemma 4.1. Let σ be a representation of G/H on H and ζ∈ H. If T is a unitary operator which intertwines σ, then WT ζ = Wζ T ∗ . Proof. We have 1 (WT ζ x)(gH) = √ cζ 1 = √ cζ 1 = √ cζ
ρ(e) ρ(g) ρ(e) ρ(g) ρ(e) ρ(g)
1/2 x, σ(gH)T ζ 1/2 x, T σ(gH)ζ 1/2
T ∗ x, σ(gH)ζ
= (Wζ oT ∗ x)(gH). Proposition 4.3. With notations as above, let K be an invariant subspace of H with projection PK . Then (i) If ζ is admissible, so is PK ζ. (ii) If ζ is bounded, so is PK ζ. 1250038-15
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Proof. If ζ be an admissible vector then we get, ρ(e) |PK ζ, σ(gH)PK ζ|2 dµ(gH) ρ(g) G/H ρ(e) ∗ |ζ, PK = σ(gH)PK ζ|2 dµ(gH) G/H ρ(g) ρ(e) ∗ |ζ, PK = PK σ(gH)ζ|2 dµ(gH) G/H ρ(g) ρ(e) |ζ, σ(gH)ζ|2 dµ(gH) < ∞. = G/H ρ(g) So the proof of (i) is complete. The second part is a consequence of Lemma 4.1.
5. Examples Let K and H be two locally compact groups with the identity elements eK , eH respectively. If there exists a homomorphism h → τh of H to Aut(K), the group of automorphisms of K, then the set K × H under operations: (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 τh1(k2 )), (h, k)−1 = (h−1 , τh−1(k −1 )), is a locally compact group, denoted by H ×τ K, is called the semidirect product of H and K. It is known that G/H K and L2 (G/H) is isometrically isomorphic to L2 (K).15 The continuous wavelet transform on semidirect product of groups are studied in Refs. 3, 9 and 12 which are special examples of continuous wavelet transform on homogeneous spaces. Example 5.1. Let N be an even number and G be the group with two generators a and b satisfying |a| = N, |b| = 3 and ba = ab2 i.e. G = a, b; aN = b3 = e, ba = ab2 . Also, let H = b, K = a and σ : G/H → U (L2 (G/H)),
σ(ai bjH)ψ(anH) = ψ(an−iH)
be a representation for all 0 ≤ i, n ≤ N − 1 and 0 ≤ j ≤ 2. It is easy to check that σ admits an admissible vector. Therefore we can define a continuous wavelet transform N −1 1 ϕ(anH)ψ(an−i H), (Wψ ϕ)(ai bjH) = √ cψ n=0 for all 0 ≤ n ≤ N − 1 and admissible vector ψ in L2 (G/H). Note that, in this example G = H ×τ K since K is not a normal subgroup of G. But H is a closed normal subgroup of G and this causes Wψ ϕ to be constant on left cosets of H.16 1250038-16
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Example 5.2. Denote by SOo (3, 1) the connected component of Lorentz group. It is a nonabelian group which may be realized as the set of all real 4 × 4 pseudoorthogonal matrices A, i.e. matrices with the property AT ηA = η, det A = 1, A00 ≥ 1, η = diag(−1, 1, 1, 1). Let So(n), the group of rotations around the origin o (3,1) be identified with two-sphere S 2 , which is not a group. It is of Rn and SO So(2) well known that SOo (3, 1) = KAH in which K SO(3), A SOo (1, 1) R SOo (3,1) SO(3)A, then every element R+ ∗ , H C (Iwasawa decomposition). Since H SOo (3,1) gH ∈ can be represented by gH ≡ (γ, a) where γ ∈ SO(3), a ∈ A. Now, H SOo (3,1) 2 2 define the representation σ of on L (S ) as: H σ:
SOo (3, 1) → U (L2 (S 2 )), H
(σ(gH)f )(ξ) = λ(γ · a, ξ)1/2 f((γ · a)−1 ξ),
for f ∈ L2 (S 2 ), ξ ∈ S 2 , where λ(γ · a, ξ) is Radon Nikodym derivative. This representation is square integrable (see details in Ref. 2). Let f ∈ L2 (S 2 ) be an admissible wavelet. Then the continuous wavelet transform is as follows: Wf (g)(γ, a) = g, σ(γ, a)f g(ξ)λ(γ · a, ξ)1/2 f((γ · a)−1 ξ)dµ(ξ) =
S2
2π π
= 0
0
g(θ, φ)λ(γ · a, (θ, φ)1/2 f((γ · a)−1 (θ, φ)) sin θdθdφ,
for g ∈ L2 (S 2 ). Also we can reconstruct g ∈ L2 (S 2 ) by 1 g= g, σ(γ, a)f σ(γ, a)f dυ(γ, a), cf SOoH(3,1) where dυ(γ, a) =
dµ(γ)da a3
and dµ(γ) is the Haar measure on SO(3).
Example 5.3. Consider Euclidean group G = SO(n) ×τ Rn with group operations (R1 , p1 ) · (R2 , p2 ) = (R1 R2 , R1 p2 + p1 ),
(R, p)−1 = (R−1 , −R−1 p).
Put n = 2 in G, i.e. G = SO(2) ×τ R2 and H = L2 (S 1 ) L2 [−π, π]. In this setting any R ∈ SO(2) and s ∈ S 1 are given explicitly by cos θ sin θ R= , −sin θ cos θ sin γ s= . cos γ The representation σ on G/H, in which H = {(0, 0, p2) ∈ G}, is defined as σ(θ, p1 )ψ(γ) = eip1 sin γ ψ(γ − θ), for all (θ, p1 ) ∈ G/H, ψ ∈ L2 (S 1 ). As has been shown in Ref. 6 σ is square integrable and so Wψ is an isometry in which ψ is an admissible wavelet. 1250038-17
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References 1. S. T. Ali, J.-P. Antoine and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations (Springer-Verlag, New York, 2000). 2. S. T. Ali, J.-P. Antoine, P. Vandergheynst and R. Murenzi, Two Dimensional Wavelets and their Relatives (Springer-Verlag, New York, 2004). 3. A. A. Arefijamaal and R. A. Kamyabi-Gol, A characterization of integrable reprerentation associated with CWT, J. Sci. Islamic Republic Iran 18 (2007) 159–166. 4. I. Bogdanova, Wavelets on non-Euclidean manifolds, Ph.D. thesis, EPFL, 2005. 5. E. Christopher, E. Hell and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989) 628–666. 6. S. Dahlke, G. Steidl and G. Teschke, Weighted coorbit spaces and Banach frames on homogeneous spaces, J. Fourier Anal. and Appl. 10 (2004) 507–539. 7. I. Daubechies, Ten Lectures on Wavelets, SIAM (1992). 8. R. Fabec and G. Olafsson, The continuous wavelet transform and symmetric spaces, Acta Appl. Math. 77 (2003) 41–69. 9. M. Fashandi, R. A. Kamyabi-Gol, A. Niknam and M. A. Pourabdollah, Continuous wavelet transform on a special homogeneous space, J. Math. Phys. (2003). 10. J. M. G. Fell and R. S. Doran, Representation of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1 (Academic Press, 1988). 11. G. B. Folland, A Course in Abstract Harmonic Analysis (CRC Press, 1995). 12. H. F¨ uhr, Continuous wavelet transform from semidirect product: Cyclic representation and plancheral measure, J. Fourier Anal. Appl. 8 (2002) 375–397. 13. H. F¨ uhr, Abstract harmonic analysis of continuous wavelet transform, Springer Lecture Notes in Mathematics (No. 1863, Berlin, 2005). 14. A. Grossmann, J. Morlet and T. Paul, Transform associated to square integrable group representation I, J. Math. Phys. 26 (1985) 2479. 15. R. A. Kamyabi-Gol and N. Tavallaei, Convolution and homogeneous spaces, Bull. Iranian Math. Soc. 35 (2009) 129–146. 16. R. A. Kamyabi-Gol and N. Tavallaei, Wavelet transforms via generalized quasi-regular representations, Appl. Comput. Harmon. Anal. 26 (2009) 291–300. 17. M. Kyed, Square integrable representations and the continuous wavelet transformation, Ph.D. Thesis, 1999. 18. H. Reiter and J. Stegeman, Classical Harmonic Analysis and Locally Compact Group (Clarendon Press, 2000). 19. Y. Sun, Y. Chen and H. Feng, Two dimensional stationary dyadic wavelet transform, decimated dyadic discrete wavelet transform and the face recognition application, Int. J. Wavelets, Multiresolut. Inf. Process. 9 (2011) 397–416. 20. T. Zhang and Q. Fan, Wavelet characterization of dyadic BMO norm and its application in image decomposition for distinguishing between texture and noise, Int. J. Wavelets, Multiresolut. Inf. Process. 9 (2011) 445–457.
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International Journal of Wavelets, Multiresolution and Information Processing Vol. 10, No. 4 (2012) 1250038 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0219691312500385
ON THE CONTINUOUS WAVELET TRANSFORM ON HOMOGENEOUS SPACES
F. ESMAEELZADEH∗,‡ , R. A. KAMYABI GOL†,§ and R. RAISI TOUSI∗,¶ ∗Department
of Pure Mathematics, Ferdowsi University of Mashhad P. O. Box 1159-91775, Mashhad, Iran
†Department
of Pure Mathematics, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS) P. O. Box 1159-91775, Mashhad, Iran ‡[email protected] §[email protected] ¶[email protected] Received 6 April 2011 Revised 8 November 2011 Published 14 July 2012
Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space H. The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto. Keywords: Homogeneous space; square integrable representation; admissible wavelet; continuous wavelet transform; reproducing kernel Hilbert space. AMS Subject Classification: Primary 43A15; Secondary 43A85, 65T60
1. Introduction and Preliminaries The wavelet transform has been an alternative to time frequency analysis. Like time frequency, wavelet transform analysis is a prominent tool in mathematics, physics, applied sciences, signal and data processing.19,20 The continuous wavelet transform on Rd is studied by several authors such as Refs. 4, 5, 7, 8 and 17 and generalized to locally compact groups and semidirect product groups by Refs. 1, 3, 9, 12–14. Ali, Antoine and Gazeau in Ref. 1 have studied continuous wavelet transform on a homogeneous space G/H using Borel sections, where G is a locally compact group and H is a closed subgroup of G, assuming a G-invariant measure on G/H (see Ref. 1, Sec. 9.2). In this paper we study continuous wavelet transform on a 1250038-1
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homogeneous space G/H where G is a locally compact group and H is a compact subgroup of G. We do this task with a completely different approach considering a relatively invariant measure on G/H. Let G be a locally compact group and H be a closed subgroup of G. Consider G/H as a homogeneous space on which G acts from the left and µ as a Radon measure on it (for a detailed account of homogeneous spaces, the reader is referred to Refs. 10, 11 and 18. For g ∈ G and Borel subset E of G/H, we define the translation µg of µ by µg (E) = µ(gE). A measure µ is said to be G-invariant if µg = µ, for all g ∈ G. The measure µ is said to be strongly quasi-invariant provided that a continuous function λ : G × G/H → (0, ∞) exists which satisfies dµg (kH) = λ(g, kH)dµ(kH), for all g, k ∈ G. If the function λ(g, ·) reduces to constant, then µ is called relatively invariant under G. A rho-function for the pair (G, H) is defined to be a continuous function ρ : G → (0, ∞) which satisfies ρ(gh) =
∆H (h) ρ(g) (g ∈ G, h ∈ H), ∆G (h)
where ∆G , ∆H are the modular functions on G and H, respectively. It is well known11 : any pair (G, H) admits a rho-function and for each rho-function ρ there is a strongly quasi-invariant measure µ on G/H such that ρ(gk) dµg (kH) = dµ ρ(k)
(g, k ∈ G).
Moreover, the existence of a homomorphism rho-function for the pair (G, H) is a necessary and sufficient condition for the existence of a relatively invariant measure on G/H. Let µ be a relatively invariant measure on G/H which arises from a rho-function ρ. It has been shown in Ref. 15 ρ(gk) =
ρ(g)ρ(k) ρ(e)
and ρ(g) dµg (kH) = , dµ ρ(e) for all g, k ∈ G. If H is a compact subgroup of G then ∆G is a homomorphism rho-function and hence G/H admits a relatively invariant measure µ which arises from a ρ-function. In the sequel we shall consider the space Lp (G/H) (1 ≤ p ≤ ∞) with respect to this measure. In this paper we need to define the basic concepts in the theory of unitary representation of homogeneous spaces. A continuous unitary representation of the homogeneous space G/H is a map σ from G/H into the group U (H), of all unitary operators on some nonzero Hilbert space H, for which gH → σ(gH)x, y is 1250038-2
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continuous from G/H into C, for each x, y ∈ H and σ(gkH) = σ(gH)σ(kH),
σ(g −1H) = σ(gH)∗ ,
for each g, k ∈ G. This defines a continuous unitary representation π of G in which the subgroup H is considered to be contained in kernel of π. Conversely, any continuous unitary representation π of G which is trivial on H induces a continuous unitary representation σ of G/H, by letting σ(gH) = π(g). Moreover, a closed subspace M of H is said to be invariant with respect to σ if σ(gH)M ⊆ M , for all g ∈ G. A continuous unitary representation σ is said to be irreducible if the only invariant subspaces of H are {0} and H. (In the sequel we always mean by a representation, a continuous unitary representation.) Let G be a locally compact group and H be a compact subgroup of G. Then as mentioned above, G/H possesses a relatively invariant measure µ. The basic example of a representation of G which arises from the natural left action of G on G/H and we call it the left type regular representation, is defined as 1/2 ρ(e) Lg ψ, (1.1) : G → U (L2 (G/H)), (g)ψ = ρ(g) where Lg ψ(kH) = ψ(g −1 kH) and L2 (G/H) is L2 (G/H, µ), where µ is a relatively invariant measure on G/H which arises from a rho-function ρ. In this paper we study the continuous wavelet transform on G/H and give a reconstruction formula for it. Among other things we show that the continuous wavelet transform is onto if and only if L2 (G/H) is a reproducing kernel Hilbert space. Our construction is useful since it is a general approach to the continuous wavelet transform on homogeneous spaces. It includes most of examples on locally compact groups, semidirect product groups as special cases and an example of a homogeneous space which is not a semidirect product of groups (see examples in Sec. 5). This paper is organized as follows: In Sec. 2 we define a square integrable representation of G/H and prove the reconstruction formula for it. As a result, we show that the set of admissible vectors is path connected (even the set of admissible vectors is a subspace). Section 3 is devoted to introducing the continuous wavelet transform on G/H. We prove that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we show that the continuous wavelet transform is onto if and only if L2 (G/H) is a reproducing kernel Hilbert space. Some properties of continuous wavelet transform on G/H is discussed in Sec. 4. It is shown that the continuous wavelet transform is a filter and so a multiplier. Finally, in Sec. 5 we present some examples which reveal the application of our results. 2. Square Integrable Representation of G/H Throughout this paper we assume that G is a locally compact group and H is a compact subgroup of G. Let H be a Hilbert space and σ be an irreducible representation 1250038-3
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of G/H on H. We aim to define a square integrable representation of G/H, an admissible vector and an admissible wavelet for such a representation. Definition 2.1. Let G be a locally compact group and H be a compact subgroup of G. An irreducible representation σ of G/H on H is said to be square integrable if there exists a nonzero element ζ ∈ H such that ρ(e) |ζ, σ(gH)ζ|2 dµ(gH) < ∞, (2.1) ρ(g) G/H where µ is a relatively invariant measure on G/H which arises from a rho-function ρ : G → (0, ∞). If ζ satisfies (2.1), it is called an admissible vector. An admissible vector ζ ∈ H is called an admissible wavelet if ζ = 1. In this case, we define the wavelet constant cζ as cζ := G/H
ρ(e) |ζ, σ(gH)ζ|2 dµ(gH). ρ(g)
(2.2)
We call cζ the wavelet constant associated to the admissible wavelet ζ. Remark 2.1. Note that since H is a compact subgroup of G, ∆G |H = ∆H = 1. So, for every rho-function ρ, we have ρ(gh) = ρ(g) for all g ∈ G and h ∈ H. This implies that there is a function ρ˜ on G/H such that for each g ∈ G, ρ˜(gH) = ρ(g). Therefore Definition 2.1 is well defined. We now prove a reconstruction formula for square integrable representation σ of G/H. To this end, we need an auxiliary lemma (Lemma 2.3) in which, we use a straightforward generalization of the classical Schur’s lemma (see Ref. 1, Lemma 4.3.1) as follows: Lemma 2.1 (Classical Schur’s lemma for homogeneous spaces). Let G be a locally compact group and H be a closed subgroup of G. Let H be a Hilbert space and σ be an irreducible representation of G/H on H. If T is a bounded operator on H which commutes with σ(gH), for all g ∈ G, i.e. T σ(gH) = σ(gH)T, then T = λI, for λ ∈ C. Similarly, it is easily shown that the extended Schur’s lemma (see Ref. 1, Lemma 4.3.3) is generalized to the homogeneous space setting as follows: Lemma 2.2 (Extended Schur’s lemma for homogeneous spaces). Let σ1 be an irreducible representation of G/H on a Hilbert space H1 and σ2 be a representation of G/H on a Hilbert space H2 . Let T : H1 → H2 be a densely defined closed linear map and the domain T is invariant under σ1 , and suppose that T intertwines σ1 and σ2 , i.e. T σ1 (gH) = σ2 (gH)T . Then T is either null or a multiple of an isometry. 1250038-4
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Lemma 2.3. Let σ be a square integrable representation of G/H, and ζ be an admissible vector for σ. Then ρ(e) |x, σ(gH)ζ|2 dµ(gH) < ∞ (2.3) M = x ∈ H, G/H ρ(g) is a closed subspace of H. Proof. Let ζ be an admissible vector and M be defined as in (2.3). Then M is invariant under σ. Indeed for all x ∈ M and k ∈ G we have ρ(e) |σ(kH)x, σ(gH)ζ|2 dµ(gH) ρ(g) G/H ρ(e) |x, σ(k −1 gH)ζ|2 dµ(gH) = ρ(g) G/H ρ(e) |x, σ(gH)ζ|2 dµk (gH) = ρ(kg) G/H ρ(e) |x, σ(gH)ζ|2 dµ(gH) < ∞. = ρ(g) G/H Since M is invariant so is M . By the irreducibility of σ we have M = H. Define the linear operator 1/2 ρ(e) x, σ(gH)ζ. (2.4) Aζ : M → L2 (G/H), Aζ (x)(gH) = ρ(g) We prove that Aζ is a closed operator. To this end, take a sequence {xn }n∈N ⊆ M, converging to x ∈ H such that Aζ xn converges in L2 (G/H) to an element ϕ ∈ L2 (G/H). Then there exists a subsequence of Aζ xn , again denoted by Aζ xn , which converges to ϕ a.e. on G/H. By the continuity of the inner product in H we have 1/2 1/2 ρ(e) ρ(e) xn , σ(gH)ζ → x, σ(gH)ζ, ρ(g) ρ(g) 1/2 ρ(e) x, σ(gH)ζ = ϕ(gH). Moreover, for all g ∈ G. Thus ρ(g) Aζ σ(gH) = (g)Aζ , where is a left type regular representation as defined in (1.1). Indeed, 1/2 ρ(e) (Aζ σ(gH)x)(kH) = σ(gH)x, σ(kH)ζ ρ(k) 1/2 ρ(e) = x, σ(g −1kH)ζ ρ(k) 1250038-5
(2.5)
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= = =
ρ(e) ρ(g) ρ(e) ρ(g) ρ(e) ρ(g)
1/2 1/2
ρ(e) ρ(g −1 k)
1/2
x, σ(g −1kH)ζ
(Aζ x)(g −1kH)
1/2 Lg (Aζ x)(kH)
= (g)(Aζ x)(kH). Now M equipped with the norm · ζ , defined as x 2ζ = x 2 + Aζ x 2L2 (G/H) , is a Hilbert space. So extended Schur’s lemma implies that Aζ from M into L2 (G/H) is a multiple of an isometry. That is there exists a positive constant λ such that, Aζ x 2L2 (G/H) = λ x 2ζ = λ x 2 + λ Aζ x 2L2 (G/H) . Hence Aζ x 2L2 (G/H) = λ 2 1−λ x , for all x ∈ M and λ < 1. Since M is dense in H we can extend ζ from H into L2 (G/H). Aζ : M → L2 (G/H) to a bounded linear operator A Let {xn }n∈N ⊆ M such that xn → x in H as n → ∞. Then Aζ xn → A˜ζ x in
L2 (G/H) as n → ∞. Since Aζ is a closed linear operator from H into L2 (G/H) with domain M , it follows that x ∈ M , i.e. M is a closed subspace of H.
Now we can state our main theorem in this section, which is the reconstruction formula. Theorem 2.1. Let σ be a square integrable representation of G/H on H. If ζ is an admissible wavelet for σ, then ρ(e) 1 x, σ(gH)ζσ(gH)ζ, ydµ(gH), (2.6) x, y = cζ G/H ρ(g) where cζ is as in (2.2). Proof. Let M be as in (2.3). By Lemma 2.3 M is a closed subspace of H. As has been shown in the proof of the lemma, M is invariant under σ and by irreducibility of σ, it follows that M = H. So Aζ is a bounded linear operator on H. Now A∗ζ Aζ intertwines with σ i.e. σ(gH)A∗ζ Aζ = A∗ζ Aζ σ(gH). Indeed, A∗ζ Aζ σ(gH)x, y = A∗ζ (g)Aζ x, y = (g)Aζ x, Aζ y = Aζ x, (g −1 )Aζ y = Aζ x, Aζ σ(g −1 H)y = σ(gH)A∗ζ Aζ x, y, where we have used (2.5) in the above equalities. By Lemma 2.1 there exists a constant c such that A∗ζ Aζ = cI, where I is the identity operator on H. Therefore 1250038-6
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we get cx, y = A∗ζ Aζ x, y = Aζ x, Aζ y Aζ x(gH)Aζ y(gH)dµ(gH) = G/H
=
G/H
ρ(e) x, σ(gH)ζσ(gH)ζ, ydµ(gH), ρ(g)
which is the desired identity. If we put x = y = ζ then ρ(e) c= |ζ, σ(gH)ζ|2 dµ(gH) = cζ . G/H ρ(g) As an interesting consequence of Theorem 2.1 we conclude that the set of admissible vectors is path connected. Corollary 2.1. The set of admissible vectors is path connected. Proof. Let ζ and ξ be two admissible vectors in H. By the following calculations, we get ζ + ξ is an admissible vector: ρ(e) |ζ + ξ, σ(gH)(ζ + ξ)|2 dµ(gH) ρ(g) G/H ρ(e) |ζ, σ(gH)ζ|2 dµ(gH) ≤ ρ(g) G/H ρ(e) |ζ, σ(gH)ξ|2 dµ(gH) + ρ(g) G/H ρ(e) |ξ, σ(gH)ζ|2 dµ(gH) + ρ(g) G/H ρ(e) |ξ, σ(gH)ξ|2 dµ(gH) + ρ(g) G/H ρ(e) |ζ, σ(gH)ζξ, σ(gH)ζ|dµ(gH) +2 ρ(g) G/H ρ(e) + |ζ, σ(gH)ζζ, σ(gH)ξ|dµ(gH) G/H ρ(g) ρ(e) |ζ, σ(gH)ζξ, σ(gH)ξ|dµ(gH) + G/H ρ(g) ρ(e) + |ξ, σ(gH)ζζ, σ(gH)ξ|dµ(gH) G/H ρ(g) 1250038-7
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+ G/H
+ G/H
ρ(e) |ξ, σ(gH)ζξ, σ(gH)ξ|dµ(gH) ρ(g)
ρ(e) |ζ, σ(gH)ξξ, σ(gH)ξ|dµ(gH) . ρ(g)
Since ξ is an admissible vector, Theorem 2.1 implies that ρ(e) |ζ, σ(gH)ξ|2 dµ(gH) ρ(g) G/H ρ(e) ζ, σ(gH)ξσ(gH)ξ, ζdµ(gH) = ρ(g) G/H = cζ ζ, ζ = cζ ζ 2 < ∞. Similarly, from the admissibility of ζ, we have ρ(e) |ξ, σ(gH)ζ|2 dµ(gH) < ∞. ρ(g) G/H Also by Schawrz inequality we get ρ(e) |ζ, σ(gH)ζξ, σ(gH)ζ|dµ(gH) G/H ρ(g) 1/2 ρ(e) 2 |ζ, σ(gH)ζ| dµ(gH) ≤ G/H ρ(g) × G/H
1/2 ρ(e) 2 |ξ, σ(gH)ζ| dµ(gH) < ∞. ρ(g)
So G/H
ρ(e) |ζ + ξ, σ(gH)(ζ + ξ)|2 dµ(gH) < ∞. ρ(g)
Therefor ζ + ξ is an admissible vector. In particular αζ + (1 − α)ξ is an admissible vector, which completes the proof. As an immediate consequence of Corollary 2.1, the set of admissible vectors is a linear space. In fact, we have the following corollary. Corollary 2.2. With notations as above, the set of all admissible vectors is a subspace of H. 1250038-8
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On the Continuous Wavelet Transform on Homogeneous Spaces
3. The Continuous Wavelet Transform on G/H In this section our notation will be the same as before. We define the continuous wavelet transform on the homogeneous space G/H and show that the range of continuous wavelet transform on G/H is a reproducing kernel Hilbert space. Also, it is shown that continuous wavelet transform is onto if L2 (G/H) is a reproducing kernel Hilbert space. Definition 3.1. Let σ be a representation of G/H on a Hilbert space H and ζ be an admissible wavelet for σ. We define the continuous wavelet transform associated to the admissible wavelet ζ as the linear operator Wζ : H → C(G/H) defined by 1/2 ρ(e) 1 x, σ(gH)ζ, (Wζ x)(gH) = √ cζ ρ(g) for all x ∈ H, g ∈ G where cζ is the wavelet constant associated to ζ as in (2.2). Also ζ ∈ H is called bounded if Wζ is bounded. Note that if σ is a square integrable representation of G/H on H and ζ is an admissible wavelet for σ, then Wζ is a bounded linear operator from H into L2 (G/H). As an application of Theorem 2.1 we can easily conclude that Wζ is an isometry. Proposition 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then (Wζ x)(gH)(Wζ y)(gH)dµ(gH), (3.1) x, y = G/H
for every x, y ∈ H. In particular, the wavelet transform Wζ : H → L2 (G/H) is an isometry. Proof. By Theorem 2.1 the proof is obvious. As a consequence of Proposition 3.1 we conclude that the range of the continuous wavelet transform is a Hilbert space. Proposition 3.2. Let σ be a square integrable representation of G/H on H. Then the range of continuous wavelet transform Wζ : H → L2 (G/H) is a Hilbert space. Proof. Denote by Rang(Wζ ) the range of Wζ . Let {ϕj }∞ j=1 ⊆ Rang(Wζ ) be such ⊆ H such that Wζ xj = ϕj , that ϕj → ϕ in L2 (G/H) as j → ∞. Choose {xj }∞ j=1 for j = 1, 2, . . . . By Proposition 3.1 we have xj − xk = Wζ xj − Wζ xk L2 (G/H) = ϕj − ϕk L2 (G/H) → 0,
j, k = 1, 2, . . . .
Therefore {xj }∞ j=1 is a Cauchy sequence in H. So xj → x for some x ∈ H as j → ∞. Since Wζ is bounded, Wζ xj → Wζ x in L2 (G/H). Thus ϕ = Wζ x, i.e. ϕ ∈ Rang(Wζ ) and the proof is complete. 1250038-9
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At this point we recall that a reproducing kernel on a measure space (B, µ) is a function k from B × B to C such that (i) k(a, a) > 0, (ii) k(a, b) = k(b, a), (iii) k(a, c) = k(a, τ )k(τ, c)dµ(τ ),
(3.2) (3.3) (3.4)
B
for all a, b, c ∈ B. A Hilbert space H of functions on a measure space B is called a reproducing kernel Hilbert space if there exists a reproducing kernel k such that x(b)k(a, b)dµ(b), (3.5) x(a) = B
for all x ∈ H, a ∈ B. We are going to show that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. In fact, we show first in the following lemma that the function pζ : G/H × G/H → C defined by pζ (gH, kH) :=
ρ(e) pζ (g −1 kH) ρ(g)
is a reproducing kernel on G/H, where 1/2 1 ρ(e) pζ (gH) = σ(gH)ζ, ζ, cζ ρ(g)
(3.6)
(3.7)
for all ζ ∈ H, g ∈ G. In Theorem 3.1 we will prove that condition (3.5) is satisfied for Rang(Wζ ). To prepare for that we need the following lemma. Lemma 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then pζ defined as in (3.6) is a reproducing kernel on the homogeneous space G/H. Proof. Condition (3.2) obviously holds, i.e. p˜ζ (gH, gH) > 0. For (3.3), the following calculations show that p˜ζ (gH, kH) = p˜ζ (kH, gH). Indeed, 1/2 ρ(e) 1 ρ(e) · p˜ζ (gH, kH) = · σ(kH)ζ, σ(gH)ζ cζ ρ(g) ρ(g−1k) 1 1/2 1 ρ(e) 2 ρ(e) σ(kH)ζ, σ(gH)ζ = cζ ρ(g) ρ(k) 1/2 ρ(e) 1 ρ(e) = σ(gH)ζ, σ(kH)ζ cζ ρ(k) ρ(k −1 g) = p˜ζ (kH, gH). 1250038-10
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Moreover, by Theorem 2.1, pζ satisfies (3.4). Indeed, p˜ζ (gH, tH)˜ pζ (tH, kH)dµ(tH) G/H
ρ(e) ρ(e) pζ (g −1tH) pζ (t−1kH)dµ(tH) ρ(g) ρ(t) G/H 1/2 ρ(e) ρ(e) 1 = 2 cζ G/H ρ(gt) ρ(g −1 t) 1/2 ρ(e) × σ(tH)ζ, σ(gH)ζσ(kH)ζ, σ(tH)ζdµ(tH) ρ(t−1 k) 1/2 1 ρ(e) ρ(e) = cζ ρ(g) ρ(g −1 k) 1 ρ(e) σ(tH)ζ, σ(gH)ζσ(kH)ζ, σ(tH)ζdµ(tH) × c G/H ζ ρ(t) 1/2 1 ρ(e) ρ(e) = σ(kH)ζ, σ(gH)ζ cζ ρ(g) ρ(g −1 k) =
=
ρ(e) pζ (g −1kH) ρ(g)
= p˜ζ (gH, kH). Theorem 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then Rang(Wζ ) is a reproducing kernel Hilbert space with reproducing kernel p˜ζ as in (3.6) such that, ψ(gH) = ψ(kH)˜ pζ (gH, kH)dµ(kH). (3.8) G/H
Proof. By Lemma 3.1 it is enough to show (3.8). Let ψ ∈ Rang(Wζ ). Then there exists an element x in H such that ψ = Wζ x. Using Theorem 2.1 we get, ψ(kH)˜ pζ (gH, kH)dµ(kH) G/H
=
ρ(e) ρ(g)
ψ(kH)pζ (g −1 kH)dµ(kH) G/H
ρ(e) ψ(gkH)pζ (kH)dµg (kH) ρ(g) G/H 1/2 ρ(e) 1 = (Wζ x)(gkH)σ(kH)ζ, ζdµ(kH) cζ G/H ρ(k)
=
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=
=
1 3/2 cζ
1 1/2 cζ
G/H
ρ(e) ρ(g)
ρ(e) ρ(g)
1/2
1/2
ρ(e) x, σ(gkH)ζσ(kH)ζ, ζdµ(kH) ρ(k)
σ(g −1 H)x, ζ
= (Wζ x)(gH) = ψ(gH). In the sequel we intend to establish a condition under which Wζ is onto. Theorem 3.2. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. If ψ ∈ L2 (G/H) and ρ(e) ψ(kH)pζ (g −1kH)dµ(kH), ψ(gH) = ρ(g) G/H where pζ for g ∈ G is as defined in (3.7) then ψ ∈ Rang(Wζ ). Proof. For every ψ ∈ L2 (G/H), the expression 1/2 1 ρ(e) ψ(gH)σ(gH)ζdµ(gH) √ cζ G/H ρ(g) defines a vector in H and the integral is weakly convergent. To prove it, note that for any x ∈ H the function 1/2 ρ(e) gH → x, σ(gH)ζ ρ(g) is in L2 (G/H) and since ψ ∈ L2 (G/H), we have the integral
1/2
1 ρ(e)
ψ(gH)x, σ(gH)ζdµ(gH)
√
cζ G/H ρ(g)
= Wζ x(gH)ψ(gH)dµ(gH)
G/H
≤ Wζ x L2 (G/H) ψ L2 (G/H) . Then by the Riesz theorem,13 the integral 1/2 1 ρ(e) ψ(gH)σ(gH)ζdµ(gH) √ cζ G/H ρ(g) ρ(e) defines a vector in H. Now, if ψ(gH) = ρ(g) ψ(kH)pζ (g −1 kH)dµ(kH), then G/H 1/2 1 ρ(e) ψ(gH)σ(gH)ζdµ(gH) (kH) = ψ(kH). (3.9) Wζ √ cζ G/H ρ(g) 1250038-12
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On the Continuous Wavelet Transform on Homogeneous Spaces
Indeed, the following calculations complete the proof: 1/2 1 ρ(e) Wζ √ ψ(gH)σ(gH)ζdµ(gH) (kH) cζ G/H ρ(g) =
1 cζ
G/H
ρ(e) ρ(g)
1/2
ρ(e) ρ(k)
1/2 ψ(gH)σ(gH)ζ, σ(kH)ζdµ(gH)
1/2 ρ(e) ψ(gH)σ(k −1gH)ζ, ζdµ(gH) ρ(gk) G/H 1/2 ρ(e) 1 ρ(e) = ψ(gH)σ(k −1gH)ζ, ζdµ(gH) cζ ρ(k) G/H ρ(k −1 g) ρ(e) = ψ(gH)pζ (k −1gH)dµ(gH) ρ(k) G/H
=
1 cζ
= ψ(kH). Corollary 3.1. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. The continuous wavelet transform Wζ is onto if and only if L2 (G/H) is reproducing kernel Hilbert space. Proof. By Theorems 3.2, 3.1 the proof is obvious. Corollary 3.2. Let σ be a square integrable representation of G/H on H and ζ be an admissible wavelet for σ. Then 1/2 ρ(e) 1 ∗ ψ(gH)σ(gH)ζdµ(gH), (3.10) Wζ (ψ) = √ cζ G/H ρ(g) for all ψ ∈ Dom(Wζ−1 ). Proof. Noticing Wζ is a unitary operator on its range, we have Wζ∗ = Wζ−1 . Now (3.9) implies the equality (3.10). 4. Some Properties of the Continuous Wavelet Transform on G/H As before let G be a locally compact group and H be a compact subgroup of G. If ζ is an admissible wavelet for the left type regular representation of G defined as in (1.1), then the continuous wavelet transform is a filter on L2 (G/H). That is, it intertwines with the left type regular representation. Proposition 4.1. Let be the left type regular representation of G on L2 (G/H) and ζ be an admissible wavelet for . Then the continuous wavelet transform Wζ : L2 (G/H) → L2 (G/H) is a filter. 1250038-13
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Proof. By the definition of continuous wavelet transform and left type regular representation of G/H we get, 1/2 ρ(e) 1 (Wζ (k)ψ)(gH) = √ (k)ψ, (g)ζ cζ ρ(g) 1/2 ρ(e) 1 = √ ψ, (k −1 g)ζ cζ ρ(g) 1/2 1/2 ρ(e) ρ(e) 1 = √ ψ, (k −1 g)ζ cζ ρ(k) ρ(k −1 g) 1/2 ρ(e) = Wζ ψ(k −1gH) ρ(k) 1/2 ρ(e) = Lk Wζ ψ(gH) ρ(k) = ((k)Wζ ψ)(gH), for all ψ ∈ L2 (G/H), k, g ∈ G. Since the continuous wavelet transform is a filter, now we intend to show that the continuous wavelet transform is a multiplier. First we need to mention Lp (G/H) is a Banach left L1 (G)-module under following action: For all 1 ≤ p ≤ ∞, there exists a left module action of L1 (G) on Lp (G/H) defined by f (k) 1/p f ∗p ψ(gH) = ρ(e) ψ(k −1gH)dk, 1/p G ρ(k) where f ∈ L1 (G) and ψ ∈ Lp (G/H). This action is well defined and under which Lp (G/H), 1 ≤ p ≤ ∞, is a Banach left L1 (G)-module with an approximate identity.15 Proposition 4.2. Let be the left type regular representation of G on L2 (G/H) and ζ be an admissible wavelet for . Then for all f ∈ L1 (G), ψ ∈ L2 (G/H) Wζ (f ∗2 ψ) = f ∗2 Wζ (ψ). Proof. Using Fubini’s theorem we get Wζ (f ∗2 ψ)(gH) 1/2 ρ(e) 1 f ∗2 ψ, (g)ζ =√ cζ ρ(g) 1/2 ρ(e) 1 (f ∗2 ψ)(kH)(g)ζ(kH)dµ(kH) =√ cζ ρ(g) G/H 1250038-14
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On the Continuous Wavelet Transform on Homogeneous Spaces
1 =√ cζ
ρ(e) ρ(g)
1/2 G/H
G
ρ(e) ρ(g)
1/2
ρ(e) ρ(t)
1/2
× f (t)ψ(t−1kH)ζ(g −1 kH)dtdµ(kH) 1 =√ cζ 1 =√ cζ
G
G
G/H
=
G
ρ(e) ρ(t)
ρ(e) ρ(g)
1/2
ρ(e) ρ(g)
1/2
1/2
ρ(e) ρ(t)
ρ(e) ρ(t)
1/2 f (t)Lt ψ(kH)(g)ζ(kH)dµ(kH)dt
1/2 f (t)Lt ψ, (g)ζdt
f (t)Wζ ψ(t−1gH)dt
= f ∗2 Wζ (ψ)(gH), for each f ∈ L1 (G), ψ ∈ L2 (G/H). Next, we show that admissibility and boundedness of an element ζ ∈ H is invariant under orthogonal projections. For showing this first we need the following lemma: Lemma 4.1. Let σ be a representation of G/H on H and ζ∈ H. If T is a unitary operator which intertwines σ, then WT ζ = Wζ T ∗ . Proof. We have 1 (WT ζ x)(gH) = √ cζ 1 = √ cζ 1 = √ cζ
ρ(e) ρ(g) ρ(e) ρ(g) ρ(e) ρ(g)
1/2 x, σ(gH)T ζ 1/2 x, T σ(gH)ζ 1/2
T ∗ x, σ(gH)ζ
= (Wζ oT ∗ x)(gH). Proposition 4.3. With notations as above, let K be an invariant subspace of H with projection PK . Then (i) If ζ is admissible, so is PK ζ. (ii) If ζ is bounded, so is PK ζ. 1250038-15
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Proof. If ζ be an admissible vector then we get, ρ(e) |PK ζ, σ(gH)PK ζ|2 dµ(gH) ρ(g) G/H ρ(e) ∗ |ζ, PK = σ(gH)PK ζ|2 dµ(gH) G/H ρ(g) ρ(e) ∗ |ζ, PK = PK σ(gH)ζ|2 dµ(gH) G/H ρ(g) ρ(e) |ζ, σ(gH)ζ|2 dµ(gH) < ∞. = G/H ρ(g) So the proof of (i) is complete. The second part is a consequence of Lemma 4.1.
5. Examples Let K and H be two locally compact groups with the identity elements eK , eH respectively. If there exists a homomorphism h → τh of H to Aut(K), the group of automorphisms of K, then the set K × H under operations: (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 τh1(k2 )), (h, k)−1 = (h−1 , τh−1(k −1 )), is a locally compact group, denoted by H ×τ K, is called the semidirect product of H and K. It is known that G/H K and L2 (G/H) is isometrically isomorphic to L2 (K).15 The continuous wavelet transform on semidirect product of groups are studied in Refs. 3, 9 and 12 which are special examples of continuous wavelet transform on homogeneous spaces. Example 5.1. Let N be an even number and G be the group with two generators a and b satisfying |a| = N, |b| = 3 and ba = ab2 i.e. G = a, b; aN = b3 = e, ba = ab2 . Also, let H = b, K = a and σ : G/H → U (L2 (G/H)),
σ(ai bjH)ψ(anH) = ψ(an−iH)
be a representation for all 0 ≤ i, n ≤ N − 1 and 0 ≤ j ≤ 2. It is easy to check that σ admits an admissible vector. Therefore we can define a continuous wavelet transform N −1 1 ϕ(anH)ψ(an−i H), (Wψ ϕ)(ai bjH) = √ cψ n=0 for all 0 ≤ n ≤ N − 1 and admissible vector ψ in L2 (G/H). Note that, in this example G = H ×τ K since K is not a normal subgroup of G. But H is a closed normal subgroup of G and this causes Wψ ϕ to be constant on left cosets of H.16 1250038-16
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On the Continuous Wavelet Transform on Homogeneous Spaces
Example 5.2. Denote by SOo (3, 1) the connected component of Lorentz group. It is a nonabelian group which may be realized as the set of all real 4 × 4 pseudoorthogonal matrices A, i.e. matrices with the property AT ηA = η, det A = 1, A00 ≥ 1, η = diag(−1, 1, 1, 1). Let So(n), the group of rotations around the origin o (3,1) be identified with two-sphere S 2 , which is not a group. It is of Rn and SO So(2) well known that SOo (3, 1) = KAH in which K SO(3), A SOo (1, 1) R SOo (3,1) SO(3)A, then every element R+ ∗ , H C (Iwasawa decomposition). Since H SOo (3,1) gH ∈ can be represented by gH ≡ (γ, a) where γ ∈ SO(3), a ∈ A. Now, H SOo (3,1) 2 2 define the representation σ of on L (S ) as: H σ:
SOo (3, 1) → U (L2 (S 2 )), H
(σ(gH)f )(ξ) = λ(γ · a, ξ)1/2 f((γ · a)−1 ξ),
for f ∈ L2 (S 2 ), ξ ∈ S 2 , where λ(γ · a, ξ) is Radon Nikodym derivative. This representation is square integrable (see details in Ref. 2). Let f ∈ L2 (S 2 ) be an admissible wavelet. Then the continuous wavelet transform is as follows: Wf (g)(γ, a) = g, σ(γ, a)f g(ξ)λ(γ · a, ξ)1/2 f((γ · a)−1 ξ)dµ(ξ) =
S2
2π π
= 0
0
g(θ, φ)λ(γ · a, (θ, φ)1/2 f((γ · a)−1 (θ, φ)) sin θdθdφ,
for g ∈ L2 (S 2 ). Also we can reconstruct g ∈ L2 (S 2 ) by 1 g= g, σ(γ, a)f σ(γ, a)f dυ(γ, a), cf SOoH(3,1) where dυ(γ, a) =
dµ(γ)da a3
and dµ(γ) is the Haar measure on SO(3).
Example 5.3. Consider Euclidean group G = SO(n) ×τ Rn with group operations (R1 , p1 ) · (R2 , p2 ) = (R1 R2 , R1 p2 + p1 ),
(R, p)−1 = (R−1 , −R−1 p).
Put n = 2 in G, i.e. G = SO(2) ×τ R2 and H = L2 (S 1 ) L2 [−π, π]. In this setting any R ∈ SO(2) and s ∈ S 1 are given explicitly by cos θ sin θ R= , −sin θ cos θ sin γ s= . cos γ The representation σ on G/H, in which H = {(0, 0, p2) ∈ G}, is defined as σ(θ, p1 )ψ(γ) = eip1 sin γ ψ(γ − θ), for all (θ, p1 ) ∈ G/H, ψ ∈ L2 (S 1 ). As has been shown in Ref. 6 σ is square integrable and so Wψ is an isometry in which ψ is an admissible wavelet. 1250038-17
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References 1. S. T. Ali, J.-P. Antoine and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations (Springer-Verlag, New York, 2000). 2. S. T. Ali, J.-P. Antoine, P. Vandergheynst and R. Murenzi, Two Dimensional Wavelets and their Relatives (Springer-Verlag, New York, 2004). 3. A. A. Arefijamaal and R. A. Kamyabi-Gol, A characterization of integrable reprerentation associated with CWT, J. Sci. Islamic Republic Iran 18 (2007) 159–166. 4. I. Bogdanova, Wavelets on non-Euclidean manifolds, Ph.D. thesis, EPFL, 2005. 5. E. Christopher, E. Hell and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989) 628–666. 6. S. Dahlke, G. Steidl and G. Teschke, Weighted coorbit spaces and Banach frames on homogeneous spaces, J. Fourier Anal. and Appl. 10 (2004) 507–539. 7. I. Daubechies, Ten Lectures on Wavelets, SIAM (1992). 8. R. Fabec and G. Olafsson, The continuous wavelet transform and symmetric spaces, Acta Appl. Math. 77 (2003) 41–69. 9. M. Fashandi, R. A. Kamyabi-Gol, A. Niknam and M. A. Pourabdollah, Continuous wavelet transform on a special homogeneous space, J. Math. Phys. (2003). 10. J. M. G. Fell and R. S. Doran, Representation of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1 (Academic Press, 1988). 11. G. B. Folland, A Course in Abstract Harmonic Analysis (CRC Press, 1995). 12. H. F¨ uhr, Continuous wavelet transform from semidirect product: Cyclic representation and plancheral measure, J. Fourier Anal. Appl. 8 (2002) 375–397. 13. H. F¨ uhr, Abstract harmonic analysis of continuous wavelet transform, Springer Lecture Notes in Mathematics (No. 1863, Berlin, 2005). 14. A. Grossmann, J. Morlet and T. Paul, Transform associated to square integrable group representation I, J. Math. Phys. 26 (1985) 2479. 15. R. A. Kamyabi-Gol and N. Tavallaei, Convolution and homogeneous spaces, Bull. Iranian Math. Soc. 35 (2009) 129–146. 16. R. A. Kamyabi-Gol and N. Tavallaei, Wavelet transforms via generalized quasi-regular representations, Appl. Comput. Harmon. Anal. 26 (2009) 291–300. 17. M. Kyed, Square integrable representations and the continuous wavelet transformation, Ph.D. Thesis, 1999. 18. H. Reiter and J. Stegeman, Classical Harmonic Analysis and Locally Compact Group (Clarendon Press, 2000). 19. Y. Sun, Y. Chen and H. Feng, Two dimensional stationary dyadic wavelet transform, decimated dyadic discrete wavelet transform and the face recognition application, Int. J. Wavelets, Multiresolut. Inf. Process. 9 (2011) 397–416. 20. T. Zhang and Q. Fan, Wavelet characterization of dyadic BMO norm and its application in image decomposition for distinguishing between texture and noise, Int. J. Wavelets, Multiresolut. Inf. Process. 9 (2011) 445–457.
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