Wandering \(r\)-tuples for unitary systems
A wandering \(r\)-tuple system that the author studies is a set \(\mathcal U\) of unitary operators on a Hilbert space \(\mathcal H\) that includes the identity and an \(r\)-tuple \((x_1,\dots,x_r)\in \mathcal H^r\) such that \(\{Ux_1,\dots, Ux_r: U\in \mathcal U\}\) is an orthonormal basis for \(\mathcal H\). The example this concept generalizes is multiwavelets in \(\mathcal H=L^2(\mathbb{R})\) where the set \(\mathcal U\) consists of operators of the form \(D^nT^m\), \(n,m\in Z\) where \((Df)(t)= \sqrt 2f(2t)\) and \((Tf)(t)= f(t-1)\). The properties of all wandering \(r\)-tuples with respect to \(\mathcal U\) are investigated using a parameterization in terms of a fixed wandering \(r\)-tuple \(\Psi=(x_1,\dots,x_r)\) and the set of all unitary operators which locally commute with \(\mathcal U\) at \(\Psi\), i.e., \(\{A\in B(\mathcal H: (AU-UA)x_i=0, U\in \mathcal U, i=1,\dots,r\}\).
scientific article; zbMATH DE number 5817890