Wavelet transform on compact Gelfand pairs and its discretization (Q1307266)

Let \((G,K)\) be a compact Gelfand pair. We know that the left regular representation of \(G\) on \(L^2(G/K)\) is decomposed into a direct sum of unitary irreducible representations \(L^2(G/K)= \oplus_{a\in \Lambda} {\mathcal H}_a\) with \(\dim{\mathcal H}_a= d_a<\infty\), where \(a_0\) is the trivial representation. A wavelet on the compact symmetric space \(G/K\) is a family \(\{\psi_\rho\}_{\rho\in(0,\infty)}\) of functions satisfying the conditions: \(\widehat \psi_\rho(a_0) =0\), \(\int^\infty_0|\widehat\psi_\rho (a)|^2 d\rho/\rho=1\) for \(a\in\Lambda \setminus\{a_0\}\) and \(\sum_{a\in \Lambda \setminus \{a_0\}} d_a \int^\infty_0 \widehat\psi_\rho (a)|^2 d\rho/\rho<\infty\), where \(^\wedge\) denotes the spherical Fourier transform on \(G/K\). The authors first define the continuous wavelet transform, prove the reconstruction formula for it and study the corresponding continuous multiresolution analysis of \(L^2 (G/K) \). They then exhibit the wavelets associated with the \(K\)-biinvariant Poisson kernel on the symmetric spaces \(SO(q)/SO(q-1)\) and \(U(q)/(q-1)\) as examples of the above general theory. From the continuous wavelet \(\{\psi_\rho\}_{\rho\in (0, \infty)}\) the authors define three types of wavelet packets and establish corresponding wavelet packet transformations, prove reconstruction formulas, and study discrete multiresolution analysis by means of these wavelet packets on \(G/K\).