Phase Space, Wavelet Transform and Toeplitz-Hankel Type Operators

Let IG(n) be the Euclidean group with dilations. It has a maximal compact subgroup SO(n−1). The homogeneous space can be realized as the phase space IG(n)/SO(n−1)≅R n ×R n . The square-integrable representation gives the admissible wavelets AW and wavelet transforms onL 2(R n ). With Laguerre polynomials and surface spherical harmonics an orthogonal decomposition of AW is given; it turns to give a complete orthogonal decomposition of theL 2-space on the phase spaceL 2(R n ×R n ,dxdy/|y| n+1) of the form ⊕ k=0 ∞ ⊕ l=0 ∞ ⊕ j=0 al A l,j k . The Schatten-von Neumann properties of the Toeplitz-Hankel type operators between these decomposition components are established.