Laugesen, Richard S. Translational averaging for completeness, characterization and oversampling of wavelets. (English) Zbl 1035.42035 Collect. Math. 53, No. 3, 211-249 (2002). Let \(\psi\in L^2(\mathbb R^{d})\) be a function. Let \(a\) and \(b\) be two \(d\times d\) invertible matrices such that \(a\) is expanding or else \(a\) is amplifying for \(\psi\). In this paper the author shows that the system \(\{ | \det a| ^{j/2} \psi(a^jx-bk) : j\in \mathbb{Z}, k\in \mathbb{Z}^d\}\) is an orthonormal basis for \(L^2(\mathbb{R}^d)\) if and only if it is orthonormal and \(\sum_{j\in\mathbb{Z}} | \hat \psi(\xi a^j)| ^2=| \det b| \) for almost every row vector \(\xi\in \mathbb{R}^d\). Other results on oversampled wavelet systems and pairs of dual frames are also discussed. Reviewer: Bin Han (Edmonton) Cited in 10 Documents MSC: 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis Keywords:orthonormal wavelet; tight frame; oversampling; dual frame PDFBibTeX XMLCite \textit{R. S. Laugesen}, Collect. Math. 53, No. 3, 211--249 (2002; Zbl 1035.42035) Full Text: EuDML