Affine systems in \(L_ 2(\mathbb{R}^d)\): The analysis of the analysis operator
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)].
scientific article; zbMATH DE number 1069301
scientific article