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Translational averaging for completeness, characterization and oversampling of wavelets. (English) Zbl 1035.42035

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)

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
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