Admissible wavelets associated with the transform group on \(\mathbb{R}^ n\) (Q1377188)

Let \(\Gamma \) be the set of diagonal matrices \(\text{diag}\{\rho _{1,\dots,}\rho _n\},\rho _i>0.\) Let \[ \mathbf P=\{(b,\rho):b\in {\mathbf R}^n,\rho \in \Gamma\}. \] Then \({\mathbf P}\) acts on \({\mathbf R}^n\) by \[ x\rightarrow (b,\rho)x=\rho x+b,\text{ where }x\in {\mathbf R}^n. \] The group law of \({\mathbf P}\) is defined by \[ (b,\rho)(b',\rho')=(\rho b'+b,\rho \rho '). \] \({\mathbf P}\) is a locally compact group with left Haar measure \(d\mu =\frac{ d^n\rho d^nb}{(\text{det } \rho )^2}.\) The authors define the unitary representation \(U\) of \({\mathbf P}\) on \(L^2({\mathbf R}^n)\) by \[ U(b,\rho )f(x)=(\text{det } \rho )^{-1/2}f(\rho ^{-1}(x-b)), \] where \(f(x)\in L^2({\mathbf R}^n),(b,\rho )\in {\mathbf P}\). The authors decompose \(L^2({\mathbf R}^n)\) into the direct sum of irreducible invariant closed subspaces under \(U\). The restrictions of \(U\) on these subspaces are square-integrable; they also give an admissibility condition in terms of the Fourier transform.