Wavelet Transforms and Symmetric Tube Domains

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 TΩ 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 TΩ . P has a natural unitary representation π on L(V ). In §2, we decompose L(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 -space on the tube domain TΩ 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 352 Liu 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 ∆(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 = c . The only eigenvalues of L(c) are 1, 12 , and 0. The corresponding eigenspaces are denoted by V (c, 1), V (c, 1 2 ) and V (c, 0). We fix a Jordan frame {c1, · · · , cr} , where r is the rank of V . Then we have the Peirce decomposition V = ⊕ j≤k Vjk where Vii = V (ci, 1) = Rci, Vij = V (ci, 1 2 ) ∩ V (cj , 1 2 ). d = dimVij , which does not depend on i and j , is called the degree of V . Let P (x) = 2L(x) − L(x) be the quadratic representation, and write x y = L(xy)− [L(x), L(y)]. For given j and for z ∈rk=j+1 Vjk the Frobenius transform τ(z) is defined by τ(z) = exp(2z cj). 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 = r ∑ j=1 ajcj , aj > 0}, N = {τ(z) · · · τ(z(r−1)) : z ∈ r ⊕ k=j+1 Vjk} are compact, diagonal and strict triangular respectively. A normalizes N and (1.1) P (a)τ(z) = τ(z̃)P (a) where z = ∑ j<k zjk, zjk ∈ Vjk, z̃ = ∑ j<k z̃jk, z̃jk ∈ Vjk, z̃jk = ak aj zjk. Liu 353 T = NA is a semi-direct product. We will use another parametrization of the triangular subgroup T . Set V+ = {u = r ∑