\(KA\)-wavelets on semisimple Lie groups and quasi-orthogonality of matrix coefficients (Q2751520)

This paper first briefly reviews the history of continuous wavelet transforms, pointing out that it is nothing but the square-integrable representation of a locally compact group \(G\) on a certain Hilbert space. Let \(G\) be a noncompact semisimple Lie group and \(G=KAK\) the Cartan decomposition of \(G\), where \(K\) and \(A\) are, respectively, the maximal and Abelian subgroups of \(G\). The author constructs an extended wavelet transform for a locally compact group of the form \(S{\times}V\), where \(S=KA\) and \(V\) is a representation space of \(G\). He explicitly presents this construction in the case of \(G=SU(1,1):\)NEWLINENEWLINENEWLINELet \((T_h, H_h)\) be the holomorphic discrete series of \(SU(1,1)\) realized on the weighted Bergman space \(H_h\) on the unit disk \(D=G/K\), and \((T_{1/2}, H_{1/2})\) the limit of the holomorphic discrete series of \(G\) realized on the Hardy space \(H_{1/2}\) on \(D\). Then \(e_n(z)=z^n\) \((n\in\mathbb N)\) is an orthonormal basis of \(H_{1/2}\). If \(\psi\) is a finite linear combination of \(e_{n+2}-e_n\), then there exists \(c\), \(0<c<1\), such that for all \(f\) in \(H_{1/2},\) NEWLINE\[NEWLINE0<c\|f\|^2{\leq}\iint_{KA}|{\langle}f, T_{1/2}(ka_t){\psi}{\rangle}|^{1/2} \sinh 2t dk dt \leq ||f ||^2/c.NEWLINE\]NEWLINE If \(\psi\) is a suitable linear combination of \(e_{n+2}-e_n\) for \(0{\leq}n{\leq}N\), then for any \(f\) in the \(L^2\)- span of \(\{e_n , n{\geq}N+1\},\) NEWLINE\[NEWLINEf(x)=c\iint_{KA}{\langle}f, T_{1/2}(ka_t){\psi}{\rangle}T_{1/2}(ka_t){\psi} \sinh 2t dk dt,NEWLINE\]NEWLINE where the constant \(c\) depends on the function \(\psi.\)NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].