Characterization of Compact Support of Fourier Transform for Orthonormal Wavelets of L2(Rd)

Let {ψμ} be an orthonormal wavelet of L2(Rd) and the support of a whole of its Fourier transform be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\bigcup\limits_\mu {{\text{supp}}{\left\{ {\hat{\psi }_{\mu } } \right\}}} } = {\prod\limits_{i = 1}^d {{\left[ {A_{i} ,D_{i} } \right]}} } - {\prod\limits_{i = 1}^d {{\left( {B_{i} ,C_{i} } \right)}} },\;\;\;\;A_{i} \leqslant B_{i} \leqslant C_{i} \leqslant D_{i} . $$\end{document}Under the weakest condition that each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left| {\hat{\psi }_{\mu } } \right|} $$\end{document}is continuous for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w \in \partial {\left( {{\prod\nolimits_{i = 1}^d {{\left[ {A_{i} ,D_{i} } \right]}} }} \right)}, $$\end{document}a characterization of the above support of a whole is given.