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Affine systems in \(L_2(\mathbb{R}^d)\). II: Dual systems. (English) Zbl 0904.42025
In part I of this paper [A. Ron and Z. Shen, J. Funct. Anal. 148, No. 2, 408-447 (1997; Zbl 0891.42018)] and in other papers of these authors, tight frames for \(L^2({\mathbb R}^d)\) were investigated that led to the construction of concrete tight wavelet frames. This paper discusses general affine and quasi-affine systems and their duals. Every affine system has a dual system, but the dual system may not be affine. However a tight affine system is its own dual. This paper concentrates on intermediate systems, i.e., affine systems whose duals are affine, too. The analysis is based on a further development of the Gramian fiberization and the mixed extension techniques introduced in previous papers. Thus the characterization of a pair of dual affine frames as being a biorthogonal Riesz basis is obtained. Also the derivation of a pair of dual affine frames from an arbitrary multiresolution analysis is described.

MSC:
42C15 General harmonic expansions, frames
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
41A63 Multidimensional problems
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