Journal of Lie Theory Volume 8 (1998) 351–366 C ◯1998 Heldermann Verlag

Wavelet Transforms and Symmetric Tube Domains

Heping Liu

Communicated by J. Faraut

Abstract. We extend wavelet analysis to the symmetric tube domains and their Shilov boundaries. Our approach is based on the theory of Jordan algebra.

One-dimensional wavelet analysis can be explained in terms of square-integrable representation of the affine group (cf. [4], [6]). It is an intermediate between the function theory on the upper half-plane of one complex variable and the harmonic analysis on the real line (cf. [7], [9]). In this paper we extend wavelet analysis to the symmetric tube domains and their Shilov boundaries, the higher dimensional analogues of the upper half-plane and the real line. We assume that V is a simple Euclidean Jordan algebra, Ω is the associated symmetric cone and T_{Ω} is the symmetric tube domain over Ω. In §1, we recall some notations and facts about Jordan algebras and symmetric cones, especially the Iwasawa subgroup P of the holomorphic automorphism group of T_{Ω} . P has a natural unitary representation π on L^{2}(V). In §2, we decompose L^{2}(V) into the direct sum of the irreducible invariant closed subspaces under π. In §3, we give an explicit characterization of the admissibility condition in terms of Fourier transform and Jordan algebra. We also give a family of admissible wavelets, which is a complete orthonormal system in a sense. Finally in §4, we use wavelet transforms to decompose the weighted L^{2} -space on the tube domain T_{Ω} into a direct sum of subspaces such that the first component is exactly the weighted Bergman space.

A good reference on Jordan algebras, symmetric cones and tube domains is the book [3] by J. Faraut and A. Korányi. Various authors developed the theory of continuous wavelet in view of square-integrable group representations, for example, in [5], [8] and in particular [1].

1. Iwasawa subgroup

Throughout this paper we keep the following assumptions and notations, which are the same as in [3].

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag

V is an n-dimensional simple Euclidean Jordan algebra with identity e. xy denotes the Jordan product of x and y. tr(x) and det(x) are defined as in [3]. We also write Δ(x) instead of det(x). The inner product on V is given by (x|y) = tr(xy). L(x) is the linear map of V defined by L(x)y = xy. An element c ∈ V is idempotent if c^{2} = c. The only eigenvalues of L(c) are 1, \frac{1}{2} , and 0. The corresponding eigenspaces are denoted by V (c, 1), V (c, \frac{1}{2}) and V (c, 0). We fix a Jordan frame {c_{1}, ⋯, c_{r}}, where r is the rank of V . Then we have the Peirce decomposition

V = ⨁_{j≤k} V_{jk}

where

V_{ii} = V (c_{i}, 1) = Rc_{i},

V_{ij} = V (c_{i}, \frac{1}{2}) ∩ V (c_{j}, \frac{1}{2}).

d = dimV_{ij}, which does not depend on i and j, is called the degree of V . Let

P (x) = 2L(x)^{2} − L(x^{2})

be the quadratic representation, and write

x□y = L(xy) − [L(x), L(y)].

For given j and for z^{(j)} ∈ ⨁_{k=j+1}^{r} V_{jk} the Frobenius transform τ(z^{(j)}) is defined by

τ(z^{(j)}) = exp(2z^{(j)}□c_{j}).

Let Ω be the symmetric cone which consists of elements x in V such that L(x) is positive definite. G(Ω) denotes the automorphism group of Ω and G is the identity component of G(Ω). G has Iwasawa decomposition G = NAK, where

K = {g ∈ G : ge = e},

A = {P (a) : a = ∑_{j=1}^{r} a_{j}c_{j}, a_{j} > 0},

N = {τ(z^{(1)}) ⋯ τ(z^{(r−1)}) : z^{(j)} ∈ ⨁_{k=j+1}^{r} V_{jk}}

are compact, diagonal and strict triangular respectively. A normalizes N and

(1.1)  P (a)τ(z^{(j)}) = τ(z̃^{(j)})P (a)

where

z^{(j)} = ∑_{j<k} z_{jk}, z_{jk} ∈ V_{jk},

z̃^{(j)} = ∑_{j<k} z̃_{jk}, z̃_{jk} ∈ V_{jk},

z̃_{jk} = \frac{a_{k}}{a_{j}} z_{jk}.

T = NA is a semi-direct product. We will use another parametrization of the triangular subgroup T. Set

V_{+} = {u = ∑_{j=1}^{r} u_{j}c_{j} + ∑_{j<k} u_{jk} : u_{j} > 0, u_{jk} ∈ V_{jk}}.

For u ∈ V_{+} , we define

t(u) = P (b_{1})τ(u^{(1)})P (b_{2}) ⋯ τ(u^{(r−1)})P (b_{r}),

where

b_{j} = c_{1} + ⋯ + c_{j−1} + u_{j}c_{j} + c_{j+1} + ⋯ + c_{r},

u^{(j)} = ∑_{k=j+1}^{r} u_{jk}.

Then

T = {t(u) : u ∈ V_{+}}.

Using (1.1), it is easy to determine the left and right Haar measures of T. The left Haar measure of T is given by

dμ_{l}(t(u)) = 2^{r} ∏_{j=1}^{r} u_{j}^{−d(j−1)−1}du,

and the right Haar measure of T is given by

dμ_{r}(t(u)) = 2^{r} ∏_{j=1}^{r} u_{j}^{−d(r−j)−1}du.

T acts simply and transitively on Ω. If

x = ∑_{j=1}^{r} x_{j}c_{j} + ∑_{j<k} x_{jk}

is the Peirce decomposition of x = t(u)e, then

x_{j} = u_{j}^{2} + \frac{1}{2} ∑_{k=1}^{j−1} ∥u_{kj}∥^{2},

x_{jk} = u_{j}u_{jk} + 2 ∑_{l=1}^{j−1} u_{lj}u_{lk}.

We identify Ω with T by identification of x = t(u)e and t(u). Then we have

dx = 2^{r} ∏_{j=1}^{r} u_{j}^{d(r−j)+1}du,

Δ(x) = ∏_{j=1}^{r} u_{j}^{2}.

Therefore

Δ(x)^{− \frac{n}{r}} dx = dμ_{l}(t(u)),

which gives the G-invariant measure on Ω.

Let V^{C} denote the complexification of V . T_{Ω} = V + iΩ is the tube domain over Ω in V^{C}. G(T_{Ω}) denotes the holomorphic automorphism group of T_{Ω} and G(T_{Ω})^{0} is the identity component of G(T_{Ω}). The Iwasawa decomposition of G(T_{Ω})^{0} is given by G(T_{Ω})^{0} = \underline{N}A\underline{K}, where

\underline{K} = \{g ∈ G(T_{Ω})^{0} : g(ie) = ie\} ⊃ K,
\underline{N} = N^{+}N,
N^{+} = {τ_{u} : z ↦ z + u, u ∈ V } ≅ V.

Therefore,

G(T_{Ω})^{0} = N^{+}T \underline{K}.

We call it the partial Iwasawa decomposition as in Terras’ book [11]. T normalizes N^{+} as

t(v)τ_{u} = τ_{t(v)u}t(v), u ∈ V, v ∈ V_{+}.

P = \underline{N}A = N^{+}T is called the Iwasawa subgroup. P is a nonunimodular group. Using the parametrization (u, v) for τ_{u}t(v) ∈ P, the left Haar measure of P is given by

dμ_{l}(u, v) = 2^{r} ∏_{j=1}^{r} v_{j}^{−d(r+j−2)−3}du dv = ∏_{j=1}^{r} v_{j}^{−d(r−1)−2}du dμ_{l}(t(v)),

and the right Haar measure of P is given by

dμ_{r}(u, v) = 2^{r} ∏_{j=1}^{r} v_{j}^{−d(r−j)−1}du dv = du dμ_{r}(t(v)).

P acts on T_{Ω} simply and transitively. We identify T_{Ω} with P by identification of τ_{u}t(v)(ie) and τ_{u}t(v). If x + iy = τ_{u}t(v)(ie) = u + it(v)e, then

Δ(y)^{− \frac{2n}{r}} dx dy = dμ_{l}(u, v),

which is the G(T_{Ω})^{0} -invariant measure on T_{Ω} . Note that

Det(g) = Δ(ge)^{\frac{n}{r}} , g ∈ G.

P has a natural unitary representation on L^{2}(V) defined by

π_{(u,v)} : f(x) ↦ Δ(t(v)e)^{− \frac{n}{2r}} f(t(v)^{−1}x − t(v)^{−1}u).

We shall decompose L^{2}(V) into the direct sum of irreducible invariant closed subspaces under π.

2. The decomposition of L^{2}(V)

In order to decompose L^{2}(V), we need to identify the non-degenerate T -orbits of V under the contragredient action of T, which is given by x ↦ t(v)′ ^{−1}x where t(v)′ denotes the transpose of t(v). First we prove

Lemma 1. (1) Suppose z_{ij} ∈ V_{ij}, w_{kl} ∈ V_{kl}, i < j, k < l, i ≠ l, k ≠ j, then

[z_{ij}□c_{i}, w_{kl}□c_{k}] = 0.

(2) Suppose z_{ij} ∈ V_{ij}, then

(z_{ij}□c_{i})′ = z_{ij}□c_{j}.

Proof. (a) To prove (1), we use the facts

V_{ij} · V_{jk} ⊂ V_{ik}, if i ≠ k,

V_{ij} · V_{kl} = {0}, if {i, j} ∩ {k, l} = Ø,

xy = \frac{1}{2}(x|y)(c_{i} + c_{j}), if x, y ∈ V_{ij}

( cf [3], Theorem IV.2.1 (iii) and Proposition IV.1.4 (i) ). We also use the matrix of z□c with respect to the Peirce decomposition, when c is idempotent in V and z ∈ V (c, \frac{1}{2}) ( see [3], proof of Lemma VI.3.1 ). Let

x = ∑_{j=1}^{r} x_{j}c_{j} + ∑_{j<k} x_{jk}, x_{jk} ∈ V_{jk}.

We compute separately in four cases.

1) If k = i, l = j, i < j, then

(z_{ij}□c_{i})(w_{ij}□c_{i})x = \frac{x_{i}}{4} (z_{ij}|w_{ij})c_{j} = (w_{ij}□c_{i})(z_{ij}□c_{i})x.

2) If k = i, l ≠ j, i < j, l, then

(z_{ij}□c_{i})(w_{il}□c_{i})x = \frac{x_{i}}{2} z_{ij}w_{il} = (w_{il}□c_{i})(z_{ij}□c_{i})x.

3) If k ≠ i, l = j, i, k < j, then

(z_{ij}□c_{i})(w_{kj}□c_{k})x = \frac{1}{2}(z_{ij}|x_{ik}w_{kj})c_{j} = \frac{1}{2}(w_{kj}|x_{ik}z_{ij})c_{j} = (w_{kj}□c_{k})(z_{ij}□c_{i})x,

where the second equality is due to the associativity of the inner product.

4) If k ≠ i, j, l ≠ i, j, i < j, k < l, we may assume i < k, then

(z_{ij}□c_{i})(w_{kl}□c_{k})x = z_{ij}(x_{ik}w_{kl}) = w_{kl}(x_{ik}z_{ij}) = (w_{kl}□c_{k})(z_{ij}□c_{i})x,

where the second equality follows from the Lemma V.3.2 in [3]. (b) Take x = z_{ij}, y = c_{i} + c_{j} in the identity

[L(x), L(y^{2})] + 2[L(y), L(xy)] = 0

( cf [3]. Proposition II.1.1 ), we obtain

[L(c_{i}), L(z_{ij})] = [L(z_{ij}), L(c_{j})].

It follows that (z_{ij}□c_{i})′ = c_{i}□z_{ij} = z_{ij}□c_{j}.

Let z_{jk} ∈ V_{jk} (j < k) and put

z^{(j)} = ∑_{k=j+1}^{r} z_{jk}, z_{(k)} = ∑_{j=1}^{k−1} z_{jk}.

Put

τ′(z_{(k)}) = exp(2z_{(k)}□c_{k}).

If z_{ij} ∈ V_{ij}, w_{kl} ∈ V_{kl}, i < j, k < l, i ≠ l, k ≠ j, Lemma 1 implies that

τ(z_{ij})τ(w_{kl}) = τ(w_{kl})τ(z_{ij})

and

τ(z_{ij})′ = τ′(z_{ij}).

Thus τ′(z_{ij}) is a dual Frobenius transform. Also, by Lemma 1,

τ(z^{(j)}) = τ(z_{j,j+1}) ⋯ τ(z_{j,r}),
τ′(z_{(k)}) = τ′(z_{1,k}) ⋯ τ′(z_{k−1,k}).

Therefore, for

u = ∑_{j=1}^{r} u_{j}c_{j} + ∑_{j<k} u_{jk}, u_{j} > 0, u_{jk} ∈ V_{jk},

we have, by also using (1.1),

t(u) = P (b_{1})τ(u^{(1)})P (b_{2}) ⋯ τ(u^{(r−1)})P (b_{r})

= P (b_{1})τ(u_{12})P (b_{2})τ(u_{13})τ(u_{23}) ⋯ P (b_{r−1})τ(u_{1r}) ⋯ τ(u_{r−1,r})P (b_{r}).

t(u)′ = P (b_{r})τ′(u_{r−1,r})τ′(u_{r−2,r}) ⋯ τ′(u_{1r})P (b_{r−1}) ⋯ P (b_{2})τ′(u_{12})P (b_{1})
= P (b_{r})τ′(u_{(r)})P (b_{r−1}) ⋯ τ′(u_{(2)})P (b_{1})

where

u_{(k)} = ∑_{j=1}^{k−1} u_{jk}.

For j = 1, ⋯, r, let V^{(j)} be the subalgebra V (c_{1} + ⋯ + c_{j}, 1) of V and W^{(j)} be the subalgebra V (c_{r−j+1} + ⋯ + c_{r}, 1) of V . P_{j} and P_{j}^{∗} denote the orthogonal projections onto V^{(j)} and W^{(j)} respectively. det_{(j)} and det_{(j)}^{∗} are the determinants relative to V^{(j)} and W^{(j)} respectively. We define

Δ_{j}(x) = det_{(j)}(P_{j}x),
Δ_{j}^{∗}(x) = det_{(j)}^{∗}(P_{j}^{∗}x).

Furthermore, for s = (s_{1}, ⋯, s_{r}). We let

Δ_{s}(x) = Δ_{1}(x)^{s_{1}−s_{2}} ⋯ Δ_{r−1}(x)^{s_{r−1}−s_{r}}Δ_{r}(x)^{s_{r}},
Δ_{s}^{∗}(x) = Δ_{1}^{∗}(x)^{s_{1}−s_{2}} ⋯ Δ_{r−1}^{∗}(x)^{s_{r−1}−s_{r}}Δ_{r}^{∗}(x)^{s_{r}}.

For x ∈ V, t(u) ∈ T, we have

(2.1)  Δ_{s}^{∗}(t(u)′x) = u_{1}^{2s_{r}} ⋯ u_{r}^{2s_{1}}Δ_{s}^{∗}(x) = Δ_{s}^{∗}(t(u)′e)Δ_{s}^{∗}(x).

In particular, Δ_{s}^{∗} is invariant under the Frobenius transform τ′(z_{(k)}) ( cf [3], Proposition VII.1.5 ).

Set

E = {ε = ∑_{j=1}^{r} ε_{j}c_{j} : ε_{j} = 1 or i},

Ω_{ε} = {x ∈ V : x = t(u)′P (ε)e, u ∈ V_{+}}.

Lemma 2. (1) The Ω_{ε} ’s are disjoint and simply transitive orbits under the contragredient action of T. (2) ⋃_{ε∈E} Ω_{ε} is a set with a complementary of measure zero.

Proof. (a) Suppose that

t(u)′P (ε)e = t(v)′P (δ)e, u, v ∈ V_{+}, ε, δ ∈ E.

Write

g = P (δ)t(v)′ ^{−1}t(u)′P (ε).

Since t(u), t(v) are triangular and P (ε), P (δ) are diagonal, g is triangular. On the other hand, since ge = e, from the Proposition VIII.2.4 in [3] g is an automorphism of V^{C} and g′ = g^{−1} . Therefore g is diagonal. Because t(u), t(v) have positive diagonal elements and P (ε), P (δ) have diagonal elements 1, −1 or i, it is concluded that u = v, ε = δ.

(b) Set

B = {x ∈ V : Δ_{k}^{∗}(x) ≠ 0, k = 1, ⋯, r}.

Obviously, V \ B is a zero measure set. We will prove that B = ⋃_{ε∈E} Ω_{ε}. It is easy to see that B ⊃ ⋃_{ε∈E} Ω_{ε}. Assume that

x = ∑_{j=1}^{r} x_{j}c_{j} + ∑_{j<k} x_{jk} ∈ B.

By [3], Theorem VI.3.5 we can write

x = τ′(z_{(r)}) ⋯ τ′(z_{(2)}) ∑_{j=1}^{r} a_{j}c_{j}

where

z_{(k)} = ∑_{j=1}^{k−1} z_{jk} ∈ ⨁_{j=1}^{k−1} V_{jk},

a_{j} = \frac{Δ_{r−j+1}^{∗}(x)}{Δ_{r−j}^{∗}(x)} ≠ 0, j = 1, ⋯, r − 1,

a_{r} = Δ_{1}^{∗}(x) ≠ 0.

Set

ε_{j} = \{ \begin{matrix}1, & if a_{j} > 0, \\ i, & if a_{j} < 0, \end{matrix}

u_{j} = \sqrt{|a_{j}|},

u_{jk} = u_{k}z_{jk}.

Then, by (1.1),

x = t(u)′P (ε)e.

Remark. Clearly, Ω_{e} = Ω, Ω_{ie} = −Ω. Ω_{ε} is a connected open set in V because Ω_{ε} is homeomorphic to V_{+} . But Ω_{ε} may not be convex neither K - invariant in general.

A simple example of Lemma 2 can be given as follows. Let V be the space Sym(m, R) of all m×m symmetric matrices and c_{j} = diag(0, ⋯, 0, 1, 0, ⋯, 0). An element t in T has the following form: tx = uxu′, where u is a lower triangular matrix with positive diagonal elements. Let Σ denote the set of all diagonal matrices with diagonal elements ±1. Ω_{σ}(σ ∈ Σ) consists of all matrices of form u′σu. Then Ω_{σ} ’s are disjoint and simply transitive orbits under the adjoint action of T and ⋃_{σ∈Σ} Ω_{σ} is a total measure set. Now we are ready to decompose L^{2}(V). Set

H_{ε} = {f ∈ L^{2}(V) : suppf̂ ⊆ Cl(Ω_{ε})}.

Proposition 1. Each of H_{ε} is an irreducible invariant closed subspace of L^{2}(V) under π and

(2.2)  L^{2}(V) = ⨁_{ε∈E} H_{ε}.

Proof. (2.2) follows from Lemma 2. Because

(π_{(u,v)}f)ˆ (y) = Δ(t(v)e)^{\frac{n}{2r}} e^{−i(u|y)}f̂(t(v)′y),

it is easy to see that H_{ε} is invariant under π. We need to prove that H_{ε} is irreducible. Let W be a non-zero invariant closed subspace of H_{ε} under π and W^{+} the orthogonal complement of W in H_{ε}. Taking a function g ∈ W, not identically zero, if f ∈ W^{+} , then

⟨f, π_{(u,v)}g⟩_{L^{2}(V)} = ∫_{V} f(x)\overline{π_{(u,v)}g(x)} dx = 0, u ∈ V, v ∈ V_{+}.

Write

g̃(x) = \overline{g(−x)},
g_{t(v)}(x) = Δ(t(v)e)^{− \frac{n}{2r}} g(t(v)^{−1}x).

We have

(2.3)  ⟨f, π_{(u,v)}g⟩_{L^{2}(V)} = f ∗ g̃_{t(v)}(u).

Therefore,

(2.4)  (f ∗ g̃_{t(v)})ˆ (y) = Δ(t(v)e)^{\frac{n}{2r}} f̂(y)\overline{ĝ(t(v)′y)} = 0, a.e. y ∈ V.

Set

S_{1} = suppf̂∩ Ω_{ε},
S_{2} = suppĝ ∩ Ω_{ε}.

S_{1}^{d} and S_{2}^{d} consist of points of density of S_{1} and S_{2} respectively. S_{2}^{d} is a positive measure set since g is not identically zero. If S_{1}^{d} has positive measure, by Lemma 2, there exists t(v_{0}) ∈ T such that S = S_{1}^{d} ∩ t(v_{0})′ ^{−1}S_{2}^{d} has positive measure. But

(f ∗ g̃_{t(v)})ˆ (y) ≠ 0, y ∈ S,

which contradicts (2.4). Therefore f is identically zero. This proves that H_{ε} is irreducible.

Remark. For F in H^{2}(T_{Ω}), the Hardy space on T_{Ω} , the following limit exists,

lim_{y→0,y∈Ω} F (· + iy) = f, in L^{2}(V).

Then

H^{2}(V) = {f ∈ L^{2}(V) : there exists F ∈ H^{2}(T_{Ω}) such that f = lim F }

is called the Hardy space on V . It is easy to see that H_{e} = H^{2}(V) and H_{ie} = \overline{H^{2}(V)}.

3. The admissibility condition

The restriction of π on H_{ε} is square-integrable, i.e., there exists a function ϕ(≠ 0) in H_{ε} such that

(3.1)  C_{ϕ} = \frac{1}{∥ϕ∥_{L^{2}(V)}^{2}} ∫_{P} |⟨ϕ, π_{(u,v)}ϕ⟩_{L^{2}(V)}|^{2}dμ_{l}(u, v) < ∞.

(3.1) is called the admissibility condition and ϕ is called an admissible wavelet. We want to give a characterization of the admissibility conditioin in terms of Fourier transform and Jordan algebra, which does not involve any group representation.

Lemma 3. Suppose x = t(u)′P (ε)e in Ω_{ε}. If

x = ∑_{j=1}^{r} x_{j}c_{j} + ∑_{j<k} x_{jk}

is the Peirce decomposition of x, then

x_{j} = ε_{j}^{2}u_{j}^{2} + \frac{1}{2} ∑_{k=j+1}^{r} ε_{k}^{2}∥u_{jk}∥^{2},

x_{jk} = ε_{k}^{2}u_{k}u_{jk} + 2 ∑_{l=k+1}^{r} ε_{l}^{2}u_{jl}u_{kl}.

Lemma 3 can be proved in a similar way as in [3], Proposition VI.3.8. For the transformation x = t(u)′P (ε)e, by Lemma 3, it is easy to

compute that

dx = 2^{r} ∏_{j=1}^{r} u_{j}^{d(j−1)+1}du

= ∏_{j=1}^{r} u_{j}^{2d(j−1)+2}dμ_{l}(t(u)).

Let

\underline{s} = (1 + d(r − 1), 1 + d(r − 2), ⋯, 1).

By (2.1),

Δ∗ ; \underline{s}(x) = Δ∗ ; \underline{s}(t(u)′e)Δ∗ ; \underline{s}(P (ε)e).

Therefore,

|Δ∗ ; \underline{s}(x)| = Δ∗ ; \underline{s}(t(u)′e).

and we have

(3.2)  |Δ∗ ; \underline{s}(x)|^{−1}dx = dμ_{l}(t(u)).

We denoted by AW_{ε} the set of all admissible wavelets in H_{ε}.

Theorem 1. Suppose ϕ(≠ 0) in H_{ε}. Then ϕ ∈ AW_{ε} if and only if

C_{ϕ} = ∫_{Ω_{ε}} |ϕ̂(y)|^{2}|Δ∗ ; \underline{s}(y)|^{−1}dy < ∞.

Proof. Using (2.3), we have

C_{ϕ} = \frac{1}{∥ϕ∥_{L^{2}(V)}^{2}} ∫_{P} |⟨ϕ, π_{(u,v)}ϕ⟩_{L^{2}(V)}|^{2}dμ_{l}(u, v)

= \frac{1}{∥ϕ∥_{L^{2}(V)}^{2}} ∫_{T} (∫_{V} |ϕ ∗ ϕ̃_{t(v)}(u)|^{2}du) ∏_{j=1}^{r} v_{j}^{−d(r−1)−2} dμ_{l}(t(v))

= \frac{1}{(2π)^{n}} \frac{1}{∥ϕ∥_{L^{2}(V)}^{2}} ∫_{T} (∫_{Ω_{ε}} |ϕ̂(y)\overline{ϕ̂(t(v)′y)}|^{2}dy) dμ_{l}(t(v))

= \frac{1}{(2π)^{n}} \frac{1}{∥ϕ∥_{L^{2}(V)}^{2}} ∫_{Ω_{ε}} |ϕ̂(y)|^{2}(∫_{T} |ϕ̂(t(v)′y)|^{2}dμ_{l}(t(v))) dy.

For y ∈ Ω_{ε}, there exists v^{1} ∈ V_{+} such that y = t(v^{1})′P (ε)e. Using (3.2) we obtain

∫_{T} |ϕ̂(t(v)′y)|^{2}dμ_{l}(t(v))
= ∫_{T} |ϕ̂((t(v^{1})t(v))′P (ε)e)|^{2}dμ_{l}(t(v))
= ∫_{T} |ϕ̂(t(v)′P (ε)e)|^{2}dμ_{l}(t(v))
= ∫_{Ω_{ε}} |ϕ̂(y)|^{2}|Δ∗ ; \underline{s}(y)|^{−1}dy.

The proof of Theorem 1 is completed.

Suppose ϕ and ψ are admissible wavelets. We define the “inner product”of ϕ and ψ by

⟨ϕ, ψ⟩_{AW} = ∫_{V} ϕ̂(y)\overline{ψ̂(y)}|Δ∗ ; \underline{s}(y)|^{−1}dy.

Remark. If ϕ ∈ AW_{ε}, ψ ∈ AW_{δ}, ε ≠ δ, then ⟨ϕ, ψ⟩_{AW} = 0. For f ∈ H_{ε}, ϕ ∈ AW_{ε}, we define the wavelet transform of f with respect to ϕ by

W_{ϕ}f(u, v) = ⟨f, π_{(u,v)}ϕ⟩_{L^{2}(V)}.

Theorem 2. Suppose f, g ∈ H_{ε}, ϕ, ψ ∈ AW_{ε}. Then

⟨W_{ϕ}f, W_{ψ}g⟩_{L^{2}(P, dμ_{l})} = ⟨ψ, ϕ⟩_{AW} ⟨f, g⟩_{L^{2}(V)}.

In particular,

∥W_{ϕ}f∥_{L^{2}(P, dμ_{l})}^{2} = C_{ϕ}∥f∥_{L^{2}(V)}^{2}.

Theorem 2 can be proved in a similar way as Theorem 1. From the theory of square-integrable representation of nonunimodular groups ( cf [2] ), Theorem 1 and Theorem 2 are equivalent.

We are going to construct a family of admissible wavelets which is complete and orthonormal with respect to ⟨·, ·⟩_{AW} .

Let {c_{jk}^{l} : l = 1, ⋯, d} be an orthonormal basis of V_{jk}. The set of indices A is defined by

A = {α ∈ V : α = ∑_{j=1}^{r} α_{j}c_{j} + ∑_{j<k} ∑_{l=1}^{d} α_{jk}^{l}c_{jk}^{l}, α_{j}, α_{jk}^{l} are nonnegative integers}.

Let L_{m}^{(μ)}(s) be the Laguerre polynomials defined by

L_{m}^{(μ)}(s) = ∑_{j=0}^{m} \binom{m + μ}{m − j}\frac{(−s)^{j}}{j!} = \frac{1}{m!}e^{s}s^{−μ}(\frac{d}{ds})^{m}(e^{−s}s^{m+μ}), μ > −1,

and H_{m}(s) be the Hermite polynomials defined by

H_{m}(s) = ∑_{j=0}^{[m/2]}(−1)^{j} \frac{m!}{j!(m − 2j)!}(2s)^{m−2j} = (−1)^{m}e^{s^{2}}(\frac{d}{ds})^{m}(e^{−s^{2}}).

The Laguerre polynomials and the Hermite polynomials satisfy the following orthogonal relations respectively.

∫_{0}^{∞} e^{−s}s^{μ}L_{m}^{(μ)}(s)L_{k}^{(μ)}(s) ds = Γ(μ + 1)\binom{m + μ}{m}δ_{mk},

∫_{−∞}^{∞} e^{−s^{2}}H_{m}(s)H_{k}(s) ds = π^{\frac{1}{2}} 2^{m}m!δ_{mk}.

And they are complete in L^{2}(R^{+}, e^{−s}s^{μ}ds) and L^{2}(R, e^{−s^{2}}ds) respectively ( cf [10], Chapter V ). We also need following notations. For

α = ∑_{j=1}^{r} α_{j}c_{j} + ∑_{j<k} ∑_{l=1}^{d} α_{jk}^{l}c_{jk}^{l} ∈ A,

we set

|α| = ∑_{j=1}^{r} α_{j} + ∑_{j<k} ∑_{l=1}^{d} α_{jk}^{l},

α! = ∏_{j=1}^{r} α_{j}! ∏_{j<k} ∏_{l=1}^{d} α_{jk}^{l}!,

α^{0} = (α_{1}, ⋯, α_{r}),

|α^{0}| = ∑_{j=1}^{r} α_{j},

α^{0}! = ∏_{j=1}^{r} α_{j}!.

For s ∈ C^{r}, λ ∈ C, we write

s + λ = (s_{1} + λ, ⋯, s_{r} + λ).

The gamma function of the symmetric cone Ω is defined by

Γ_{Ω}(s) = ∫_{Ω} e^{−tr(x)}Δ_{s}(x)Δ(x)^{− \frac{n}{r}} dx

= (2π)^{\frac{n−r}{2}} ∏_{j=1}^{r} Γ(s_{j} − (j − 1)\frac{d}{2}), Res_{j} > (j − 1)\frac{d}{2}, j = 1, ⋯, r.

Assume that

v = ∑_{j=1}^{r} v_{j}c_{j} + ∑_{j<k} ∑_{l=1}^{d} v_{jk}^{l}c_{jk}^{l} ∈ V_{+},

we define

ψ_{α}(v) = C ∏_{j=1}^{r} e^{−v_{j}^{2}}v_{j}^{ν− \frac{n}{r}} L_{α_{j}}^{(μ_{j})}(2v_{j}^{2}) ∏_{j<k} ∏_{l=1}^{d} e^{− \frac{1}{2} (v_{jk}^{l})^{2}}H_{α_{jk}^{l}}(v_{jk}^{l})

where

C = 2^{\frac{1}{2} (νr−n+|α|−|α^{0}|)}α^{0}!(α!)^{− \frac{1}{2}} Γ_{Ω}(α^{0} + ν − \frac{n}{r})^{− \frac{1}{2}} ,

μ_{j} = ν − \frac{n}{r} − 1 − \frac{d}{2}(j − 1), ν > 1 + d(r − 1).

We regard ψ_{α}(v) as the functions on group T. From the orthogonal relations and completeness of the Laguerre polynomials and Hermite polynomials, we conclude that {ψ_{α}(v) : α ∈ A} is an orthonormal basis of L^{2}(T, dμ_{l}).

Now we define a family of functions ϕ_{α}^{ε} by

ϕ̂_{α}^{ε}(y) = \{ \begin{matrix}ψ_{α}(v), & y = t(v)′P (ε)e, \\ 0, & y ∉ Ω_{ε}. \end{matrix}

Then ϕ_{α}^{ε} ∈ H_{ε} is admissible and {ϕ_{α}^{ε} : ε ∈ E, α ∈ A} is an complete orthonormal system with respect to ⟨·, ·⟩_{AW} .

4. The decomposition of L_{ν}^{2}(T_{Ω})

The weighted L^{2} -space on the symmetric tube domain T_{Ω} is defined by

L_{ν}^{2}(T_{Ω}) = \{F : ∥F ∥_{ν}^{2} = ∫_{T_{Ω}} |F (x + iy)|^{2}Δ(y)^{ν− \frac{2n}{r}} dx dy < ∞\}.

H_{ν}^{2}(T_{Ω}), the weighted Bergman space, is the subspace of all holomorphic functions in L_{ν}^{2}(T_{Ω}), i.e.,

H_{ν}^{2}(T_{Ω}) = {F ∈ L_{ν}^{2}(T_{Ω}) : F is holomorphic on T_{Ω}}.

We assume that ν > 1 + d(r − 1) so that H_{ν}^{2}(T_{Ω}) ≠ {0}. We want to decompose L_{ν}^{2}(T_{Ω}) into the direct sum of subspaces such that the first component is exactly the weighted Bergman space H_{ν}^{2}(T_{Ω}).

Suppose f ∈ H_{ε}, ϕ ∈ AW_{ε}. For ν > 1+d(r −1), we define the weighted wavelet transform W_{ϕ}^{ν} by

F (u + it(v)e) = W_{ϕ}^{ν}f(u + it(v)e) = C_{ϕ}^{− \frac{1}{2}} W_{ϕ}f(u, v)Δ(t(v)e)^{− \frac{ν}{2}} .

Set

H_{α}^{ε} = {F = W_{ϕ_{α}^{ε}}^{ν}f : f ∈ H_{ε}}.

Proposition 2. For ν > 1 + d(r − 1), we have

(4.1)  L_{ν}^{2}(T_{Ω}) = ⨁_{ε∈E,α∈A} H_{α}^{ε}

and

H_{0}^{e} = H_{ν}^{2}(T_{Ω}).

Proof. By Theorem 2, W_{ϕ}^{ν} is an isometric operator from H_{ε} into L_{ν}^{2}(T_{Ω}) and H_{α}^{ε} ’s are mutually orthogonal subspaces of L_{ν}^{2}(T_{Ω}). We need to prove

L_{ν}^{2}(T_{Ω}) ⊂ ⨁_{ε∈E,α∈A} H_{α}^{ε}.

Suppose F ∈ L_{ν}^{2}(T_{Ω}). Write F_{v}(u) = F (u + it(v)e). For v ∈ V_{+} almost every where, F_{v} ∈ L^{2}(V). We let G(v, y) = F̂_{v}(y). Fix y = t(v^{1})′P (ε)e ∈ Ω_{ε}, then G(v, y), regarded as the function on T, is in L^{2}(T, Δ(t(v)e)^{ν− \frac{n}{r}} dμ_{l}(t(v)) ). Since

\{Δ(t(v)e)^{− \frac{ν}{2} + \frac{n}{2r}} ϕ̂_{α}^{ε}(t(v)′y) : α ∈ A\}

is an orthonormal basis of L^{2}(T, Δ(t(v)e)^{ν− \frac{n}{r}} dμ_{l}(t(v)) ), we get

G(v, y) = ∑_{α∈A} a_{α}(y)Δ(t(v)e)^{− \frac{ν}{2} + \frac{n}{2r}} ϕ̂_{α}^{ε}(t(v)′y), y ∈ Ω_{ε}.

We define the functions f_{α}^{ε} by

f̂_{α}^{ε}(y) = \{ \begin{matrix}a_{α}(y), & y ∈ Ω_{ε}, \\ 0, & y ∉ Ω_{ε}. \end{matrix}

It is easy to see that f_{α}^{ε} ∈ H_{ε}. Therefore we have

F (u + it(v)e) = ∑_{ε∈E,α∈A} W_{ϕ_{α}^{ε}}^{ν}f_{α}^{ε}(u + it(v)e).

This proves (4.1).

Now we assume that

F (u + it(v)e) = W_{ϕ_{0}^{e}}^{ν}f(u + it(v)e), f ∈ H_{e}.

Note that

tr(y) = ∑_{j=1}^{r} y_{j} = ∑_{j=1}^{r} v_{j}^{2} + \frac{1}{2} ∑_{j<k} ∑_{l=1}^{d} |v_{jk}^{l}|^{2},

Δ(y) = ∏_{j=1}^{r} v_{j}^{2}, y = t(v)′e,

we have

ϕ̂_{0}^{e}(y) = \{ \begin{matrix}Γ_{Ω}(ν − \frac{n}{r})^{− \frac{1}{2}} 2^{\frac{νr−n}{2}} e^{−try}Δ(y)^{\frac{ν}{2} − \frac{n}{2r}} , & y ∈ Ω, \\ 0, & y ∉ Ω. \end{matrix}

Therefore,

F (u + it(v)e) = Δ(t(v)e)^{− \frac{ν}{2} − \frac{n}{2r}} ∫_{V} f(x)\overline{ϕ(t(v)^{−1}x − t(v)^{−1}u)} dx

= (2π)^{−n}Δ(t(v)e)^{− \frac{ν}{2} + \frac{n}{2r}} ∫_{Ω} f̂(y)e^{i(u|y)}\overline{ϕ̂(t(v)′y)} dy

= (2π)^{−n}Γ_{Ω}(ν − \frac{n}{r})^{− \frac{1}{2}} ∫_{Ω} e^{i(u+it(v)e|y)}f̂(y)Δ(2y)^{\frac{ν}{2} − \frac{n}{2r}} dy,

where we make use of the equality

Δ(t(v)e) = Det(t(v))^{\frac{r}{n}} = Det(t(v)′)^{\frac{r}{n}} = Δ(t(v)′e).

We define the map F_{ν} by

g(y) = F_{ν}f(y) = (2π)^{− \frac{n}{2}} Γ_{Ω}(ν − \frac{n}{r})^{− \frac{1}{2}} f̂(y)Δ(2y)^{− \frac{ν}{2} + \frac{n}{2r}}

and the map L_{ν} by

F (u + it(v)e) = L_{ν}g(u + it(v)e) = (2π)^{− \frac{n}{2}} ∫_{Ω} e^{i(u+it(v)e|y)}g(y)Δ(2y)^{ν− \frac{n}{r}} dy.

It is obvious that F_{ν} is an isomorphism from H^{2}(V) onto L_{ν}^{2}(Ω) = L^{2}(Ω, Δ(2y)^{ν− \frac{n}{r}} dy). L_{ν} is an isomorphism from L_{ν}^{2}(Ω) onto H_{ν}^{2}(T_{Ω}) ( cf [3] ). We see that W_{ϕ_{0}^{e}}^{ν} = L_{ν} ∘ F_{ν} and

∥F ∥_{L_{ν}^{2}(T_{Ω})}^{2} = Γ_{Ω}(ν − \frac{n}{r})∥g∥_{L_{ν}^{2}(Ω)}^{2} = ∥f∥_{L^{2}(V)}^{2}.

Clearly, H_{0}^{e} is exactly the weighted Bergman space H_{ν}^{2}(T_{Ω}).

Acknowledgements. The author is grateful to Professor Tom H. Koornwinder for reading and revising the manuscript.

The work was carried out while the author was visiting the University of Amsterdam. The author was also supported by the National Natural Science Foundation of China.

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School of Mathematics Sciences Peking University Beijing 100871 People’s Republic of China

Received September 5, 1997 and in final form April 21, 1998