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
In light of the results of Chui and Sun @cite_23 , it is not surprising that, for integer dilations, all of the previously studied conditions on tightness preserving lattices @math are equivalent to our newly introduced condition . We state these conditions in the proposition below.
@cite_23
1983772857 2208813460 2114789426 2073656615
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) In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. We consider the problem of robustly recovering a @math -sparse coefficient vector from the Fourier series that it generates, restricted to the interval @math . The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of @math ) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm @math , we show upper and lower bounds on the minimax error rate that both scale like @math , providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity @math , or equivalently from comparing @math to the so-called @math -spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n).