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Orthogonal frames of translates. (English) Zbl 1042.42038
Summary: Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in \(L^2(\mathbb R^d)\). The characterizations are applied to Bessel sequences which have an affine structure, and a quasi-affine structure. These also lead to characterizations of superframes. Moreover, we characterize perfect reconstruction, i.e., duality, of subspace frames for translation invariant (bandlimited) subspaces of \(L^2(\mathbb R^d)\).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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