We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace. Oversampling of wavelet frames has been a subject of extensive study by several authors dating back to the early 1990s. The first oversampling results are due to Chui and Shi [16, 17], who proved that oversampling by odd factors preserves tightness of dyadic affine frames. This is now the central result of the subject known as the Second Oversampling Theorem. Its higher dimensional generalizations to integer matrix dilations were studied by Chui and Shi [18], Johnson [27], Laugesen [30], and Ron and Shen [31]. In particular, these authors introduced (in several equivalent forms) the class of oversampling matrices ‘relatively prime’ to a fixed dilation A and they established several oversampling results for (not necessarily tight) affine frames. Dutkay and Jorgensen [23] shed a new light on these results by showing that oversampling of orthonormal (or frame) wavelets by such matrices leads to orthonormal (or frame) superwavelets, respectively. Chui and Sun [22] have completed the understanding of the case of integer dilations by showing that the class of ‘relatively prime’ matrices is optimal for the Second Oversampling Theorem; that is, if an oversampling matrix falls out of this class, then the oversampling does not preserve a tight frame property in general. However, it is possible to give a characterization of oversampling matrices preserving tightness once affine frame generators are chosen. These
Laugesen [Theorem 8.3] La02 has proved the Second Oversampling Theorem for dual frames for (and expansive or amplifying) dilations @math under the assumption and, as usual, @math . Within the settings of expansive dilations, Theorem 8.3 in @cite_31 is therefore a special case of Theorem .
@cite_31
2008731398 1983772857 2034220622 2296866626
Abstract Let M be a dilation matrix, Ψ a finite family of L 2 -functions, and P the collection of all nonsingular matrices P such that M, P, and P M P −1 have integer entries. The objective of this paper is two-fold. First, for each P in P , we characterize all tight affine frames X ( Ψ , M ) generated by Ψ such that the over-sampled affine systems X P ( Ψ , M ) relative to the “over-sampling rate” P remain to be tight frames. Second, we characterize all over-sampling rates P ∈ P , such that the over-sampled affine systems X P ( Ψ , M ) are tight frames whenever the affine system X ( Ψ , M ) is a tight frame. Our second result therefore provides a general and precise formulation of the second over-sampling theorem for tight affine frames. Abstract : Discrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are 'global' in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions. (AN) Abstract The Kantorovich–Rubinstein theorem provides a formula for the Wasserstein metric W 1 on the space of regular probability Borel measures on a compact metric space. Dudley and de Acosta generalized the theorem to measures on separable metric spaces. Kellerer, using his own work on Monge–Kantorovich duality, obtained a rapid proof for Radon measures on an arbitrary metric space. The object of the present expository article is to give an account of Kellerer’s generalization of the Kantorovich–Rubinstein theorem, together with related matters. It transpires that a more elementary version of Monge–Kantorovich duality than that used by Kellerer suffices for present purposes. The fundamental relations that provide two characterizations of the Wasserstein metric are obtained directly, without the need for prior demonstration of density or duality theorems. The latter are proved, however, and used in the characterization of optimal measures and functions for the Kantorovich–Rubinstein linear programme. A formula of Dobrushin is proved. Two central limit theorems for sample Fr 'echet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first one, Theorem 2.2, encompasses and improves upon most earlier CLTs on Fr 'echet means and broadens the scope of the methodology beyond manifolds to diverse new non-Euclidean data including those on certain stratified spaces which are important in the study of phylogenetic trees. It does not require that the underlying distribution @math have a density, and applies to both intrinsic and extrinsic analysis. The second theorem, Theorem 3.3, focuses on intrinsic means on Riemannian manifolds of dimensions @math and breaks new ground by providing a broad CLT without any of the earlier restrictive support assumptions. It makes the statistically reasonable assumption of a somewhat smooth density of @math . The excluded case of dimension @math proves to be an enigma, although the first theorem does provide a CLT in this case as well under a support restriction. Theorem 3.3 immediately applies to spheres @math , @math , which are also of considerable importance in applications to axial spaces and to landmarks based image analysis, as these spaces are quotients of spheres under a Lie group @math of isometries of @math .