On characterizations of multiwavelets in \(L^{2}(\mathbb{R}^n)\) (Q2731918)

The author gives a new characterization of multiwavelets \(\{\psi^1,\psi^2,\dots, \psi^L\}\) in \(L^2(\mathbb{R}^n)\). In particular, he proves that the equation NEWLINE\[NEWLINE\sum^L_{\ell= 1}\int_{\mathbb{R}^n} |\widehat\psi^\ell(\xi)|^2 {d\xi\over \rho(\xi)}= \kappa(\rho),NEWLINE\]NEWLINE where \(\kappa(\rho)\) denotes the characteristic number of a quasi norm \(\rho\), leads to the completion of the orthonormal system \(\{\psi^\ell_{j,k}: j\in\mathbb{Z}\), \(k\in\mathbb{Z}^n\), \(\ell= 1,2,\dots,L\}\). This in turn leads to a quick proof of the completeness theorem due to \textit{G. Garrigós} and \textit{D. Speegle} [Proc. Am. Math. Soc. 128, No. 4, 1157-1166 (2000; Zbl 0969.42016)].