数理解析研究所講究録
第 1925 巻 2014 年 94-102

94

Continuous wavelet transforms on spaces ofvector-valued functions
Kazuhide Oshiro
Graduate School of Mathematics, Nagoya University
Abstract
Let be the semidirect product of a locally compact abelian group$N$ with a closed subgroup $H$ of $Aut(N)$ .
We consider continuous
wavelet transforms associated to unitary representations of realizedon spaces of vector-valued square integrable functions on $N.$
$G$

$G$

1

Introduction

Continuous wavelet transforms for the semidirect product group with a commutative normal subgroup have been studied by many authors. The simplest
example is the one associated to a quasi-regular representation of the $ax+b$group [2]. Furthermore wavelet transforms for a semidirect group with a unimodular, not necessarily commutative, normal subgroup are studied in [8].
Let be the semidirect group $N\lambda H$ of a locally compact abelian group$N$ and a closed subgroup $H$ of $Aut(N)$ .
An element $g\in G$ is written as$g=(n, h)$ with $n\in N$ and $h\in H$ . This group law is given by$G$$(n, h)(n’, h’)=(n+hn’, h’h) (n, n’\in N, h, h’\in H)$ .

Let $d\mu_{H}(h)$ denote a left Haar measure of $H$ and $dn$ a Lebesgue measure on$N$ . We define the measure ofby$G$$d\mu_{G}(g)=\delta(h)^{-1}dnd\mu_{H}(h) , g=(n, h)\in N\rangle\triangleleft H,$

is asuch that $d(hn)=\delta(h)dn$ . Then
where is the map from $H$ toleft Haar measure of . Let be an irreducible unitary representation of $H$$\delta$$\mathbb{R}_{+}$$G$$\sigma$$d\mu_{G}$95
on a Hilbert space
spaceof$L^{2}(N, \mathcal{H}_{\sigma})$$\mathcal{H}_{\sigma}$. We define the unitary representation of-valued square integrable functions on $N$ by
$\pi$

$G$

on the$\mathcal{H}_{\sigma}$$\pi(n, h)f(n_{0})=\delta(h)^{-\frac{1}{2}}\sigma(h)f(h^{-1}(n_{0}-n))$$(n, n_{0}\in N, h\in H)$ .

. In
This representation is equivalent to the induced representation
particular, when is trivial,
is called a quasi-regular representation. In
this case, continuous wavelet transforms arising from have been developedin various directions [7, 8, 11, 12, 13, 14]. In this paper, we consider a moregeneral case. We introduce the wavelet transforms obtained from the unitary
representation with not necessarily finite dimensional.
$Ind_{H}^{G}\sigma$$\pi$$\sigma$

$\pi$

$\pi$

2

$\sigma$Preliminaries

In this section, we recall basic notions about wavelet transform associatedto a unitary representation of a locally compact group. Let
be a locallycompact group and an irreducible unitary representation of defined on
a complex separable Hilbert space. The representation is said to be
if there exists a nonzero vector
such that the imageof the map:
given by
$G$

$G$

$\pi$

$\mathcal{H}_{\pi}$$\pi$$\mathcal{S}quare-integ\underline{ra}ble$$W_{\varphi}$$\varphi\in \mathcal{H}_{\pi}$$\mathcal{H}_{\pi}arrow C(G)$$\overline{W}_{\varphi}\psi(9)=\langle\psi, \pi(g)\varphi\rangle (\psi\in \mathcal{H}_{\pi}, g\in G)$

is contained in $L^{2}(G)$ , that is,$\int_{G}|\overline{W}_{\varphi}\psi(g)|^{2}d\mu(g)<\infty$

for all$\psi\in \mathcal{H}_{\pi}$. Then$\varphi$

is called an admissible vector.

Theorem 1 ([1, Theorem 3.1]). Suppose is a square integrable representation of defined on. There exists a unique positive self-adjoint operator$C$
whose domain coincides with the set of admissible vectors such that
$\pi$

$G$

$\int_{G}\langle W_{\varphi_{1}}\psi_{1}(g)$$\mathcal{H}_{\pi}$,$W_{\varphi_{2}}\psi_{2}(g)\rangle d\mu(g)=\langle\psi_{1},$for any admissible vectors$\varphi_{1}$and$\varphi_{2}.$$\psi_{2}\rangle\langle C\varphi_{2},$$C\varphi_{1}\rangle$$(g\in G, \psi_{1}, \psi_{2}\in \mathcal{H}_{\pi})$.

96
For an admissibe vector , we definein Theorem 1, we have$\varphi$$C_{\varphi}=\langle C\varphi,$$C\varphi\rangle$. Applying$\varphi_{1}=$$\varphi_{2}=\psi_{1}=\psi_{2}=\varphi$$C_{\varphi}= \frac{1}{\langle\varphi,\varphi\rangle}\int_{G}|\overline{W}_{\varphi}\varphi(g)|^{2}d\mu(g)<\infty.$

We define the map

from$W_{\varphi}$$\mathcal{H}_{\pi}$into $L^{2}(G)$ by$W_{\varphi}\psi=C_{\varphi}^{-\frac{1}{2}}\overline{W}_{\varphi}\psi (\psi\in \mathcal{H}_{\pi})$Then$W_{\varphi}$.

is isometry by Theorem 1, so that for any$\psi\in \mathcal{H}_{\pi}$

we have$\psi=\int_{G}W_{\varphi}\psi(g)\pi(g)\varphi d\mu(g)$in the weak sense. The map

3$W_{\varphi}$

is called a$continuou\mathcal{S}$wavelet transform.

Construction of the wavelet transforms associated to$\pi$

From now on, let
be the semidirect product group as in Section 1. Weby the unitary dual of $N$ . Since $N$ is commutative, any element of$N$ isone-dimensional. The dual action of on
is defined by$G$$\hat{N}$$denote\wedge$$\hat{N}$$G$$g\cdot\nu(n)=\nu(g^{-1}ng) (g\in G, \nu\in\hat{N}, n\in N)$

For each$v\in\hat{N}$, we denote by$G_{v}$.

the stabilizer of , that is,$\nu$$G_{v}=\{g\in G;g\cdot v=v\},$

which is a closed subgroup of . We define $H_{v}=G_{v}\cap H$ . Then
We denote by
the -orbit inthrough :$G$$\mathcal{O}_{v}$$G_{v}=N\rangle\triangleleft H_{v}.$$\hat{N}$

$G$

$v$

$\mathcal{O}_{\nu}=\{g\cdot v, g\in G\}.$In this section, we construct the wavelet transforms associated to the unitary
representation after giving an irreducible decomposition of
For the study of irreducible subrepresentation of , it is useful to introduce
a unitary representation which is equivalent to . We define the Fouriertransform on
by
$\pi$

$\pi.$

$\pi$

$\pi$

$\mathcal{F}$$L^{2}(N, \mathcal{H}_{\sigma})$$\mathcal{F}f(\nu)=\hat{f}(v)=\int_{N}v(n)f(n)dn (v\in N$

97
Taking the conjugate of by , we obtain the unitary representationon. The representation is described as$\pi$$\mathcal{F}\circ\pi\circ \mathcal{F}^{-1}$$\mathcal{F}$$L^{2}(\hat{N}, \mathcal{H}_{\sigma})$$\hat{\pi}=$$\hat{\pi}$(1)$\hat{\pi}(n, h)\varphi(v)=v(n)\delta(h)^{\frac{1}{2}}\sigma(h)\varphi(h^{-1}\cdot v) (\varphi\in L^{2}(\hat{N}, \mathcal{H}_{\sigma}$

Now let us assume the following [3, 8] :(A1) The orbit space is countably separated, that is, there is a countablefamilyof -invariant Borel set inissuch that each orbit in
the intersection of all the’s that contain it.
$\hat{N}$$G$$\{E_{j}\}$$\hat{N}$$\{E_{j}\}$(A2) For each $\nu\in\hat{N}$ , the map $G/G_{\nu}\ni gG_{\nu}\mapsto g$
homeomorphism.

$\nu\in \mathcal{O}_{\nu}$is a

be the Plancherel measure on . There exists elements$v_{k}(k\in K)$ of, indexed by some set $K$ , such that
and$=\emptyset(k\neq k’)$ and(A3) Let$\hat{N}$$\mu$$\hat{N}$$\mu(\mathcal{O}_{v_{k}})>0$$\mu(\hat{N}\backslash \sqcup_{k\in K}\mathcal{O}_{\nu_{k}})=0.$$\mathcal{O}_{v_{k}}\cap \mathcal{O}_{v_{k}},$(A4) The stabilizer $H_{v_{k}}=\{h\in H ; h\cdot\nu_{k}=v_{k}\}$ at eachcompact.
(A5) For every $k\in K$ , the restriction
there exists a index set
such that$\sigma|_{H_{\nu_{k}}}$$\Lambda_{k}$$v_{k}\in\hat{N}$isis multiplicity free. Namely,
and$\rho_{\alpha}\not\simeq$$\sigma|_{H_{v_{k}}}=\oplus_{\alpha\in\Lambda_{k}}\rho_{\alpha}$$\rho_{\alpha’}(\alpha\neq\alpha$We say thatis regular if the two conditions (A1) and (A2) are satisfied.
If
and is an irreducible representation of, we define a unitary
representation
of
by
$G$

$v\in\hat{N}$$H_{v}$$\rho$$v\otimes\rho$$G_{\nu}$$(v\otimes\rho)(n, h)=v(n)\rho(h) (n\in N, h\in H_{v})$ .

Theorem 2 ([3, Theorem 6.42]). Suppose
is regular. If
and isan irreducible unitary representation of , thenis an irreducible
representation of G. Every irreducible unitary representation of is equivalent to one of this form. Moreover,andare equivalent$v’=g\cdot\nu$, and $harrow\rho(h)$if and only if and belong to the same orbit, say
and $harrow\rho’(g^{-1}hg)$ are equivalent representation of$v\in\hat{N}$$G$$H_{\nu}$$\rho$$Ind_{G_{v}}^{G}v\otimes\rho$$G$$Ind_{G_{\nu}}^{G}v\otimes\rho$$v$$Ind_{G_{v’}}^{G}v’\otimes\rho’$$v’$$H_{\nu}.$

The following theorem is useful in order to investigate whether
is square-integrable.

$Ind_{G_{\nu}}^{G}\nu\otimes\rho$98Theorem 3 ([10, Theorem 2 Let
representation of. The representation$v\in\hat{N}$$H_{v}$and only ifand$\mu(\mathcal{O}_{v})>0$$\rho$

and$\rho$

be an irreducible unitary
square-integrable if$Ind_{G_{\nu}}^{G}v\otimes\rho i_{\mathcal{S}}$is square-integrable.

For $k\in K$ , we regard
extension. Thanks to (1), the spacebythe subrepresentation$L^{2}(\mathcal{O}_{\nu_{k}}, \mathcal{H}_{\sigma})$as a subspace ofby zerois -invariant. We denote. By the assumption (A3), we have$L^{2}(\hat{N}, \mathcal{H}_{\sigma})$$G$$L^{2}(\mathcal{O}_{v_{k}}, \mathcal{H}_{\sigma})$$\hat{\pi}_{k}$$\hat{\pi}|_{L^{2}(\mathcal{O}_{\nu_{k}},\mathcal{H}_{\sigma})}$$\hat{\pi}=\oplus_{k\in K}\hat{\pi}_{k}.$Proposition 1. The unitary representation$\hat{\pi}_{k}$is equivalent to$Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes$$\sigma|_{H_{\nu_{k}}}.$

Proof. We denote by

the unitary representation. Let
be the canonical quotient map from. The unitary representation
tois the left-regular representation on the Hilbert space completion
the spacedefined by$\Pi_{k}$$Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes\sigma|_{H_{v_{k}}}$

$G$

$q$

$G_{\nu_{k}}$$\tilde{\mathcal{L}}_{k,\sigma}of$$\Pi_{k}$$\mathcal{L}_{k,\sigma}$$\mathcal{L}_{k,\sigma}=\{F:Garrow \mathcal{H}_{\sigma}$

;

$q($

suppF) is compact and$F((n, h)(n’, h =v_{k}(n’)^{-1}\sigma(h’)^{-1}F(n, h)$ for$n, n’\in N, h\in H, h’\in H_{v_{k}}\}$

with the inner product$\langle F, F’\rangle=\int_{G/G_{v_{k}}}\langle F(g) , F’(9)\rangle_{\sigma}d\mu_{G/G_{v_{k}}}(gG_{v_{k}})$We define the map$\Phi$

from$\mathcal{L}_{k,\sigma}$to$L^{2}(\mathcal{O}_{\nu_{k}}, \mathcal{H}_{\sigma})$by$\Phi(F)(v)=\delta(h)^{\frac{1}{2}}\sigma(h)F(O, h) (v=h\cdot\nu_{k})$The inverse map$\Phi^{-1}$.

.

is given by$\Phi^{-1}\varphi(n, h)=\delta(h)^{-\frac{1}{2}}h\cdot v_{k}^{-1}(n)\sigma(h)^{-1}\varphi(h\cdot v_{k})$.

The map extends to a unitary operator fromonto
. There$\hat{\pi}_{k}(n,h)0\Phi=\Phi
0\Pi_{k}(n,h)$
fore, it suﬃces to show thatfor all $(n, h)\in G.$
For any, we have$\Phi$$\tilde{\mathcal{L}}_{k,\sigma}$$L^{2}(\mathcal{O}_{v_{k}}, \mathcal{H}_{\sigma})$$F\in \mathcal{L}_{k,\sigma}$$\hat{\pi}_{k}(n, h)\circ\Phi F(\nu)=v(n)\delta(h)^{\frac{1}{2}}\sigma(h)\Phi(F)(h^{-1}\cdot\nu)$.

99
On the other hand, we have$\Phi\circ\Pi_{k}(n, h)F(v)$$=$$\delta(h’)^{\frac{1}{2}}\sigma(h’)\Pi_{k}(n, h)F(O, h’)$$= \delta(h)^{\frac{1}{2}}\sigma(h’)\varphi(-h^{-1}n, h^{-1}h’)$$= \delta(h)^{\frac{1}{2}}h^{-1}h’\cdot v_{k}(h^{-1}n)\sigma(h)\Phi(F)(h^{-1}h’\cdot v_{k})$$= \delta(h)^{\frac{1}{2}}v(n)\sigma(h)\Phi(F)(h^{-1}\cdot v)$

where $v=h’\cdot v_{k}$ . Therefore we see thatintertwines$\Phi$,$\hat{\pi}_{k}$and$\square$$\Pi_{k}.$Proposition 2 ([3, Proposition 6.9]). Let $G’$ be a closed subgroup of G. If
any family of unitary representations of $G’$ , thenis equivalent$Ind_{G}^{G},$ $(\oplus\tau_{\beta})$$\{\tau_{\beta}\}i\mathcal{S}$$to\oplus Ind_{G}^{G},\tau_{\beta}.$

By Proposition 1 and Proposition 2, the unitary representationisequivalent to. Combining Theorem 2 with the remarks
following Theorem 3 and the assumption (A5), we see that is multiplicity
free and. By Theorem 3, an irreducible
unitary representation
is square-integrable by the assumption(A4) because every irreducible unitary representation of a compact group issquare-integrable. Therefore we obtain the following proposition:$\hat{\pi}_{k}$$\oplus_{\alpha\in\Lambda_{K}}Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes\rho_{\alpha}$$\hat{\pi}$$\hat{\pi}=\oplus_{k\in K}\oplus_{\alpha\in\Lambda_{K}}Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes\rho_{\alpha}$$Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes\rho_{\alpha}$

Proposition 3. Irreducible decomposition of the unitary representation
into
is multiplicity free. Moreover, for each$k\in K$ andis square, the induced representation
integrable.
$\hat{\pi}$$\oplus_{k\in K}\oplus_{\alpha\in\Lambda_{K}}Ind_{G_{v_{k}}}^{G}v_{k}\otimes\rho_{\alpha}$$\alpha\in\Lambda_{K}$$Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes\rho_{\alpha}$. By the assumpWe construct the representation space oftion (A5),is finite dimenis decomposed intoand eachsional representation on the Hilbert space. Therefore
is a direct sumof irreducible-invariant subspaces, that is,$Ind_{G_{\nu_{k}}}^{G}v_{k}\otimes\rho_{\alpha}$$\oplus_{\alpha\in\Lambda_{k}}\rho_{\alpha}$$\sigma|_{H_{\nu_{k}}}$$\rho_{\alpha}$$\mathcal{H}_{\sigma}$$\mathcal{H}_{\rho_{\alpha}}$$H_{v_{k}}$$\mathcal{H}_{\sigma}=\bigoplus_{\alpha\in\Lambda_{k}}\mathcal{H}_{\rho_{\alpha}}$We define the invariant subspace$\mathcal{L}_{k,\sigma,\alpha}$

of

(2)

.

$\mathcal{L}_{k,\sigma}$by$\mathcal{L}_{k,\sigma,\alpha}=\{\varphi\in \mathcal{L}_{k,\sigma} ; \varphi(n, h)\in \mathcal{H}_{\rho_{\alpha}}, a.a.

The Hilbert completion

By (2), the space
the subspace$\tilde{\mathcal{L}}_{k,\sigma}$$\mathcal{H}_{k,\sigma,\alpha}$$\tilde{\mathcal{L}}_{k,\sigma,\alpha}$

is the representation space ofis decomposed asof$\Phi(\tilde{\mathcal{L}}_{k,\sigma,\alpha})$(n, h)\in G\}.$$\oplus_{\alpha\in\Lambda_{K}}\tilde{\mathcal{L}}_{k,\sigma,\alpha}$$L^{2}(\mathcal{O}_{v_{k}}, \mathcal{H}_{\sigma})$.

$Ind_{G_{k}}^{G_{U}}v_{k}\otimes\rho_{\alpha}.$. Now we denote by

100
Lemma 1. For any$v\in \mathcal{O}_{\nu_{k}}$, we define$\mathcal{H}_{\alpha,\nu}=\sigma(h)\mathcal{H}_{\rho_{\alpha}},$where $v=h\cdot v_{k}(h\in H)$ . Then
described as$\mathcal{H}_{\alpha,v}$

is well-defined. Moreover$\mathcal{H}_{k,\sigma,\alpha}=\{\varphi\in L^{2}(\mathcal{O}_{v_{k}}, \mathcal{H}_{\sigma});\varphi(v)\in \mathcal{H}_{\alpha,\nu}a.a.

Proof. For any element$\Phi(F)$$\varphi\in \mathcal{H}_{k,\sigma,\alpha}$

there exists$\mathcal{H}_{k,\sigma,\alpha}$is\nu\}.$$F\in\tilde{\mathcal{L}}_{k,\sigma,\alpha}$such that$\varphi=$. Then$\varphi(\nu)=\Phi(F)(v)=\delta(h)^{\frac{1}{2}}\sigma(h)\varphi(O, h)\in\sigma(h)\mathcal{H}_{\rho_{\alpha}},$therefore we have$\mathcal{H}_{k,\sigma,\alpha}\subset\{F\in L^{2}(\mathcal{O}_{v_{k}}, \mathcal{H}_{\sigma});\varphi(\nu)\in \mathcal{H}_{\alpha,\nu}a.a.

On the other hand, for any
have$\varphi\in L^{2}(\mathcal{O}_{v_{k}}, \mathcal{H}_{\sigma})$satisfing\nu\}.$$\varphi(\nu)\in \mathcal{H}_{\alpha,v}a.a.$

$v$

, we

$\Phi^{-1}\varphi(n, h)=\delta(h)^{-\frac{1}{2}}h\cdot v_{k}(n)\sigma(h)\varphi(h\cdot v_{k})\in \mathcal{H}_{\rho_{\alpha}}.$Therefore we see that$\Phi^{-1}\varphi\in\tilde{\mathcal{L}}_{k,\sigma,\rho}$, so that

Proposition 4. Irreducible
is multiplicity free.

$decomp_{0\mathcal{S}}ition$$\square$$\varphi\in \mathcal{H}_{k,\sigma,\alpha}.$of the space$L^{2}(\hat{N}, \mathcal{H}_{\sigma})$

into$\oplus_{k\in K}\oplus_{\alpha\in\Lambda_{K}}\mathcal{H}_{k,\sigma,\alpha}$

Let us construct the wavelet transforms associated to . We choose anadmissible vector
such thatfor each and . We
assume that$\pi$$\varphi_{k,\alpha}\in \mathcal{H}_{k,\sigma,\alpha}$(A6)$\varphi=\sum_{k\in K}\sum_{\alpha\in\Lambda_{K}}\varphi_{k,\alpha}$

Theorem 4. Put
to $L^{2}(G)$ by$k$$C_{\varphi_{k,\alpha}}=1$

converge in$f=\mathcal{F}^{-1}\varphi\in L^{2}(N, \mathcal{H}_{\sigma})$$L^{2}(\hat{N}, \mathcal{H}_{\sigma})$.

. We can define the map$L^{2}(N, \mathcal{H}_{\sigma})$$W_{f}\psi(g)=\langle\psi, \pi(g)f\rangle (\psi\in L^{2}(\hat{N}, \mathcal{H}_{\sigma}$

Then$W_{f}i_{\mathcal{S}}$isometry, and for any$\psi\in L^{2}(N, \mathcal{H}_{\sigma})$$\psi=\int_{G}W_{f}\psi(g)\pi(g)fd\mu_{G}(g)$in the weak sense.

$\alpha$

we have$W_{f}$

from

101

Proof. For any$\psi=\mathcal{F}^{-1}\phi\in L^{2}(N, \mathcal{H}_{\sigma})(\phi\in L^{2}(\hat{N},$we have$\mathcal{H}_{\sigma}$$\int_{G}|W_{f}\psi(g)|^{2}d\mu_{G}(g)=\int_{G}|\langle\psi,$ $\pi(n, h)f\rangle|^{2}d\mu_{G}(g)=\int_{G}|\langle\phi,$ $\hat{\pi}(n, h)\varphi\rangle|^{2}d\mu_{G}(g)$By Proposition 4 and the orthogonality formula, the last term equals$\sum_{k\in K}\sum_{\alpha\in\Lambda_{K}}\int_{G}|\langle\phi_{k,\alpha}, \hat{\pi}(n, h)\varphi_{k,\alpha}\rangle|^{2}d\mu_{G}(g)$whereexpression above equals$\phi=\sum_{k\in K}\sum_{\alpha\in\Lambda_{K}}\phi_{k,\alpha}(\phi_{k,\alpha}\in \mathcal{H}_{k,\sigma,\alpha})$,. Theorem 1 tell us that the$\sum_{k\in K}\sum_{\alpha\in\Lambda_{K}}C_{\varphi_{k,\alpha}}\langle\phi_{k,\alpha}, \phi_{k,\alpha}\rangle=\langle\phi, \phi\rangle=\langle\psi, \psi\rangle$

since$C_{\varphi_{k,\alpha}}=1$. Therefore we have$\int_{G}|W_{f}\psi(g)|^{2}d\mu_{G}(g)=\langle\psi, \psi\rangle$

for any$\psi\in L^{2}(N, \mathcal{H}_{\sigma})$. Hence, Theorem 4 is proved.

$\square$Acknowledgements
The present author would like to express his gratitude to his advisor Professor
Hideyuki Ishi for his constant encouragement and helpful discusstion.

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Graduate School of Mathematics
Nagoya university
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