PHASE-SPACE, WAVELET TRANSFORM AND TOEPLITZ-HANKEL TYPE OPERATORS

Let IG(n) be the Euclidean group with dilations. It has a maximal comp act subgroup SO(n - 1). The homogeneous space can be realized as the p hase space IG(n)/SO(n - 1) congruent to R(n) x R(n). The square-integr able representation gives the admissible wavelets AW and wavelet trans forms on L(2)(R(n)). With Laguerre polynomials and surface spherical h armonics an orthogonal decomposition of AW is given; it turns to give a complete orthogonal decomposition of the L(2)-space on the phase spa ce L(2)(R(n) x R(n), dxdy/\y\(n+1)) of the form +(infinity)(k=0) +(inf inity)(l=0) +(al)(j=0) A(l,j)(k). The Schatten-von Neumann properties of the Toeplitz-Hankel type operators between these decomposition comp onents are established.