An abstract interpretation of the wavelet dimension function using group representations (Q1568537)

To a multiwavelet \(\psi_1, \cdots ,\psi_n\) a multiplicity function (generalizing the wavelet dimension function) is associated. This leads to necessary and sufficient conditions for a multiwavelet (with respect to an arbitrary expansive integral matrix) to be an MRA-wavelet. Among the consequences of this approach for \(\psi \in L^2(R)\) are the results (i) if \(\psi\) is an MRA-wavelet then the support \(S\) of the Fourier transform \(\widehat{\psi}\) is either equal to \(R\) or \(S\) contains no interval of length \(\geq 8\pi\), (ii) if \(\widehat{\psi}\) has compact support, then the support of \(\widehat{\psi}\) contains no interval of length \(\geq 8\pi\) (whether or not \(\psi\) is associated to an MRA).