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.


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


Zbl 0891.42018
Full Text: DOI EuDML


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