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Affine systems in \(L_ 2(\mathbb{R}^d)\): The analysis of the analysis operator. (English) Zbl 0891.42018
The generation of wavelets from a multiresolution analysis of \(L_2({\mathbb{R}}^d)\) is usually restricted to (bi)orthogonal systems which are not redundant. However, when switching to frames, one runs into difficulties. This paper gives a framework to tackle these problems. Suppose we work on lattices in \({\mathbb{R}}^d\). An affine system \(X\) consists of the orbits generated by a discrete analog of the affine group (“dilations” and “translations”) applied to a finite set of functions (“mother functions”). The problem of deciding whether such a system is a (tight) frame is approached via an in depth discussion of the analysis operator. That is the operator \(T_X^*\) which maps \(f\in L_2({\mathbb{R}}^d)\) onto the sequence of coefficients \((\langle f,x\rangle)_{x\in X}\in\ell_2(X)\) (it is the adjoint of the synthesis operator). The authors finally arrive at the construction of a tight frame via a multiresolution analysis. This paper is a continuation of the authors’ previous papers on this subject: [Can. J. Math. 47, No. 5, 1051-1094 (1995; Zbl 0838.42016) and “Weyl-Heisenberg frames and Riesz bases in \(L_2({\mathbb{R}}^d)\)”, Duke Math. J. (in press)].

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