The orthonormal dilation property for abstract Parseval wavelet frames (Q2862953)

Let \(\Gamma_0\) be a countable discrete group, \(\alpha: \Gamma_0 \rightarrow \Gamma_0\) a monomorphism, NEWLINE\[NEWLINEG(\alpha; \Gamma_0):= \langle u,\Gamma_0: u \gamma u^{-1}=\alpha(\gamma), \forall\gamma\in\Gamma_0\rangleNEWLINE\]NEWLINE \(\Gamma_1=\{u^{j}:j\in\mathbb{Z}\}\) and \(\Gamma=\Gamma_1\Gamma_0\). Let \(\pi\) be a faithful unitary representation of \(G\) on a Hilbert space \(\mathcal{H}\) and \(\psi \in \mathcal{H}\).NEWLINENEWLINEUsing positive-definite maps, a sufficient condition is obtained on the group \(G\) in order that for every Parseval wavelet frame \(\pi(\Gamma)\psi\) for \(\mathcal{H}\), there exist a Hilbert space \(\mathcal{K}\) containing \(\mathcal{H}\), \(\Psi \in \mathcal{K}\) and a faithful unitary representation \(\tau\) of \(G\) on \(\mathcal{K}\) with \(\tau(g)_{|\mathcal{H}}=\pi(g)\) for all \(g \in G\), such that \(P_{\mathcal{H}}\Psi=\psi\) and \(\pi(\Gamma)\psi\) is an orthonormal basis for \(\mathcal{K}\).NEWLINENEWLINEExamples of such groups \(G\) are presented. Some well-known function systems, including both affine wavelet systems and shearlet systems, are particular cases of these general Parseval wavelet frames \(\pi(\Gamma)\psi\).