共查询到20条相似文献,搜索用时 31 毫秒
1.
We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H. Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H. An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m,H). This classification system is applied to MRAs and other classical examples in L2(Rd) as well as to previously studied abstract examples. 相似文献
2.
Lawrence W. Baggett Nadia S. Larsen Kathy D. Merrill Judith A. Packer Iain Raeburn 《Journal of Fourier Analysis and Applications》2009,15(5):616-633
Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space ℋ that fail to be multiresolution
analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed
scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space ℋ is L
2(ℝ
n
), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition, which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity
function m. 相似文献
3.
Lawrence W. Baggett Herbert A. Medina Kathy D. Merrill 《Journal of Fourier Analysis and Applications》1999,5(6):563-573
An abstract formulation of generalized multiresolution analyses is presented, and those GMRAs that come from multiwavelets are characterized. As an application of this abstract formulation, a constructive procedure is developed, which produces all wavelet sets in n
relative to an integral expansive matrix. 相似文献
4.
《Applied and Computational Harmonic Analysis》1996,3(4):366-371
This paper is concerned with the development of an equivalence relation between two multiresolution analysis ofL2(R). The relation called unitary equivalence is created by the action of a unitary operator in such a way that the multiresolution structure and the decomposition and reconstruction algorithms remain invariant. A characterization in terms of the scaling functions of the multiresolution analysis is given. Distinct equivalence classes of multiresolution analysis are derived. Finally, we prove that B-splines give rise to nonequivalent examples. 相似文献
5.
In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations.
We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and
Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets.
In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity)
extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3
interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension
function for which the conventional algorithm for constructing integer dilated wavelet sets fails. 相似文献
6.
S. Albeverio S. Evdokimov M. Skopina 《Journal of Fourier Analysis and Applications》2010,16(5):693-714
We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions)
is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling
functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set
of wavelet functions generates a p-adic wavelet frame. 相似文献
7.
We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.
8.
The paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis (GMRA). We show the existence of a scaling vector of a GMRA such that its first component exhausts the spectrum of the core space near the origin. The corresponding low-pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector. We also prove a generalization of the Unitary Extension Principle for an infinite number of generators. This results in the construction scheme for tight framelets using low-pass and high-pass matrix masks generalizing the classical MRA constructions. We prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function. 相似文献
9.
In this second paper, we study the case of substitution tilings of
\mathbb Rd{{\mathbb R}^d} . The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j = 0, . . . , d − 1. We reconstruct the tiling’s equivalence relation in a purely combinatorial way using the AF-relations given by the lower
dimensional substitutions. We define a Bratteli multi-diagram B{{\mathcal B}} which is made of the Bratteli diagrams Bj, j=0, ?d{{\mathcal B}^j, j=0, \ldots d} , of all those substitutions. The set of infinite paths in Bd{{\mathcal B}^d} is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in Bj{{\mathcal B}^j} for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation RB{{\mathcal R}_{\mathcal B}} on B{{\mathcal B}} by saying that two infinite paths are equivalent if they have borders which are tail equivalent in Bj{{\mathcal B}^j} for some j ≤ d. We show that RB{{\mathcal R}_{\mathcal B}} is homeomorphic to the tiling’s equivalence relation RX{{\mathcal R}_\Xi} . 相似文献
10.
Fernando De Terán 《Linear and Multilinear Algebra》2013,61(12):1605-1628
We give a complete solution of the matrix equation AX?+?BX ??=?0, where A, B?∈?? m×n are two given matrices, X?∈?? n×n is an unknown matrix, and ? denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A?+?λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A?+?λB to the canonical form by strict equivalence. 相似文献
11.
The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function
of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p
−1. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable
function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there
exist infinity many different 2-adic orthonormal wavelet bases in ℒ2(ℚ2) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ2(ℚ2
n
) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral
analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also
that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary
prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators
develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).
相似文献
12.
Marilyn Breen 《Periodica Mathematica Hungarica》2007,55(2):169-176
A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that
are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every
two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path.
Communicated by Imre Bárány 相似文献
13.
The main goal of this paper is the development of the MRA theory in . We described a wide class of p-adic refinement equations generating p-adic multiresolution analyses. A method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example which gives a new 3-adic wavelet basis. Another realization leads to the p-adic Haar bases which were known before. 相似文献
14.
Directional Haar wavelet frames on triangles 总被引:3,自引:0,他引:3
Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C2-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising. 相似文献
15.
16.
S. A. Evdokimov M. A. Skopina 《Proceedings of the Steklov Institute of Mathematics》2009,266(Z1):143-154
Within the theory of multiresolution analysis, a method of constructing 2-adic wavelet systems that form Riesz bases in L
2(ℚ2) is developed. A realization of this method for some infinite family of multiresolution analyses leading to nonorthogonal
Riesz bases is presented. 相似文献
17.
A multi-bridge hypergraph is an h-uniform linear hypergraph consisting of some linear paths having common extremities. In this paper it is proved that the
multisets of path lengths of two chromatically equivalent multi-bridge hypergraphs are equal provided the multiplicities of
path lengths are bounded above by 2
h-1 − 2. Also, it is shown that h-uniform linear cycles of length m are not chromatically unique for every m, h ≥ 3. 相似文献
18.
Lawrence W. Baggett Veronika Furst Kathy D. Merrill Judith A. Packer 《Journal of Functional Analysis》2009,257(9):2760-2779
We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik. 相似文献
19.
Biswaranjan Behera 《Journal of Mathematical Analysis and Applications》2007,328(2):1237-1246
A multiresolution analysis was defined by Gabardo and Nashed for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA. Further, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form orthonormal bases for L2(R). 相似文献
20.
Maciej Paluszyński Hrvoje Šikić Guido Weiss Shaoliang Xiao 《Journal of Geometric Analysis》2001,11(2):311-342
A tight frame wavelet ψ is an L
2(ℝ) function such that {ψ jk(x)} = {2j/2
ψ(2
j
x −k), j, k ∈ ℤ},is a tight frame for L
2 (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight
frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained
from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions
and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained
by other authors. 相似文献