An Equivalence Relation on Wavelets in Higher Dimensions

Abstract

The action of translation operators on wavelet subspaces in higher dimensions is investigated. This action defines an equivalence relation on the set of single wavelets of L 2 (R n ) associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of the wavelets. Further, examples of wavelets in each of these classes are constructed. This construction shows the existence of wavelets for which the associated wavelet subspaces are invariant under various groups of translation operators.

Oversampling of wavelet frames for real dilations

“…In spite of this, we show that this class of lattices plays an important role in linking oversampling with additional shift invariance of the core space. More precisely, the preservation of the tight frame property when oversampling by such lattices is actually equivalent with the membership in Behera-Weber classes of wavelets [2,33]. Other results on the dilation matrix oversampling were obtained earlier in [9,15].…”

“…We also give a counterexample to one of the results claimed in the paper of Chui, Czaja, Maggioni, and Weiss [15]. Finally, in Section 5 we show results on the equivalence of tight frame preservation for dilation matrix oversampling of the translation lattice and the membership in Behera-Weber classes of wavelets [2,33].…”

“…Following Behera [2] and Weber [33] we define the classes of wavelets with respect to the extent of shift invariance of corresponding spaces of negative dilates. Definition 5.1.…”

Dimension Functions of Rationally Dilated GMRAs and Wavelets

“…Namely, the core space V 0 of such a GMRA enjoys more shift-invariance than the ordinary Z N -SI. We should mention here that the study of integer dilated wavelets with improved shift-invariance goes back to Weber [37], see also [9,34]. Note that in the case when A is integervalued, no such improvement exists.…”