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Affine frames, quasi-affine frames, and their duals. (English) Zbl 0892.42019
Summary: The notion of quasi-affine frame was recently introduced by A. Ron and Z. Shen [J. Funct. Anal. 148, No. 2, 408-447 (1997; Zbl 0891.42018)] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in the paper by A. Ron and Z. Shen (loc. cit.). Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems.

MSC:
42C15 General harmonic expansions, frames
47N40 Applications of operator theory in numerical analysis
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