Necessary condition and sufficient condition for affine frames

The main goal of this paper is to establish a set of necessary conditions for affine frames.These conditions are also sufficient for tight frames.     

小波框架的某些判定准则

本文首先给出了希尔伯特空间H上两个半小波框架序列成为小波框架的一个充分条件,该结论与框架的扰动理论有关.然后建立了通过膨胀与平移母函数生成L~2(R)上的小波框架的膨胀参数,平移参数以及母函数的一些充分条件.这些结果推广了小波框架理论中经典文献中的相关结论.我们还对贝塞尔序列进行了讨论,并得到了一些有趣的结论.     

Adjointness Aspects of the Down-Set Functor

The down-set construction, when applied to the category of Boolean frames, can be viewed as a functor into the category of frames with frame homomorphisms subject to various conditions akin to openness. We prove that it has a right adjoint, which is then given by Booleanization, exactly for near openness and one other, closely related property; a similar result is obtained for the finitary case of pseudocomplemented distributive lattices. In addition, we present a characterization of the frames which are down-set frames of Boolean frames.     

Optimal dual frames for erasures II

We continue the investigation on optimal dual frames for erasures. We obtain an necessary and sufficient condition under which the canonical dual frames are the unique optimal dual frames for erasures. We examine several special simple conditions under which the canonical dual is either not optimal or it is optimal dual but not unique one.     

Some New Inequalities for Wavelet Frames on Local Fields

Wavelet frames have gained considerable popularity during the past decade,primarily due to their substantiated applications in diverse and widespread fields of science and engineering.Finding general and verifiable conditions which imply that the wavelet systems are wavelet frames is among the core problems in time-frequency analysis.In this article,we establish some new inequalities for wavelet frames on local fields of positive characteristic by means of the Fourier transform.As an application,an improved version of the Li-Jiang inequality for wavelet frames on local fields is obtained.     

Pair frames in Hilbert $$\varvec{C^*}$$-modules

In this paper, we introduce pair frames in Hilbert \(C^*\)-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert \(C^*\)-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert \(C^*\)-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.     

小波紧框架的构造

小波框架理论是小波分析的重要内容之一.本文对于4-带尺度函数,由V1中的l个函数ψ1,ψ2,…,ψl构造小波紧框架.首先给出这个l个函数构成小波紧框架的充分条件.由此给出由4-带尺度函数构造出一个小波紧框架的公式.最后还给出类似于小波的小波紧框架的分解与重构算法.     

Sufficient Conditions for Shift-Invariant Systems to be Frames in L~2（R~n）

The main purpose of this paper is to establish some new sufficient conditions under which shift-invariant systems become frames in L2（Rn）. As applications, we obtain new results for Gabor frames.     

非均匀Gabor框架的稳定性

研究了当窗函数变化时非均匀Gabor框架的稳定性.对紧支撑Gabor框架,将均匀情况下关于稳定性的结论推广到了非均匀的情况;对一般的Gabor框架,利用W（L^∞,e^1）范数给出了其稳定的一个充分条件.     

Weighted fusion frame construction via spectral tetris

Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Given a spectrum for a desired fusion frame operator and dimensions for subspaces, one existing method for creating unit-weight fusion frames with these properties is the flexible and elementary procedure known as spectral tetris. Despite the extensive literature on fusion frames, until now there has been no construction of fusion frames with prescribed weights. In this paper we use spectral tetris to construct more general, arbitrarily weighted fusion frames. Moreover, we provide necessary and sufficient conditions for when a desired fusion frame can be constructed via spectral tetris.     

α带小波紧框架的显式构造方法

文中研究了对应于α-带尺度函数的小波紧框架,这个小波紧框架是由V₁中的n个函数ψ¹,ψ²,...,ψⁿ构成. 首先给出了这n个函数构成小波紧框架的充分条件, 并借助尺度函数给出了构造小波紧框架的显式公式. 如果尺度函数的符号是有理函数,则可以构造出符号为有理函数的小波紧框架. 其次给出类似于正交小波的小波紧框架的分解与重构算法,并构造了小波紧框架的数值算例.     

Super-Wavelets and Decomposable Wavelet Frames

A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given.     

K-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces.By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Frame Based Methods for Unconstrained Optimization

This paper describes a wide class of direct search methods for unconstrained optimization, which make use of fragments of grids called frames. Convergence is shown under mild conditions which allow successive frames to be rotated, translated, and scaled relative to one another.     

g-Banach框架的稳定性研究(英文)

Abdollahpour引入了g-Banach框架的概念,本文研究了g-Banach框架的扰动和稳定性,得到了关于g-Banach框架的一些稳定性条件,Abdollahpour文献中有关稳定性结果是本文结果的特例.     

Exact g-frames in Hilbert spaces

G-frames, which were considered recently as generalized frames in Hilbert spaces, have many properties similar to those of frames, but not all the properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. In this paper, we firstly give a characterization of an exact g-frame in a complex Hilbert space. We also obtain an equivalent relation between an exact g-frame and a g-Riesz basis under some conditions. Lastly we consider the stability of an exact g-frame for a Hilbert space under perturbation. These properties of exact g-frames for Hilbert spaces are not similar to those of exact frames.     

Strong Nearness Frames Having Enough Cauchy Filters

We prove results establishing sufficient conditions for the sum of two nearness frames to have enough Cauchy filters. From these results and the fact that, in the category of strong nearness frames having enough Cauchy filters and uniform frame maps, complete spatial frames form a coreflective subcategory, follow a variety of results where the open-sets contravariant functor from topological spaces to frames transforms products into sums and inverse limits into direct limits.     

Dimension invariance of finite frames of translates and Gabor frames

A dimension invariance property for finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation of duals in a very large dimensional space is now easily accessible by merely working in a space of some much smaller dimension. A number of uniform and non-uniform schemes are studied. To satisfy the support conditions, a method of finding valid alternate dual functions with small support via a known parametric dual frame formula is discussed. Oftentimes it is convenient to have truncated approximate duals that satisfy the support conditions. Stability studies of the dimension invariance principle via such approximate duals are also presented.     

Equiangular tight frames from complex Seidel matrices containing cube roots of unity

We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity. We also exhibit a relationship between these equiangular tight frames, complex Seidel matrices, and highly regular, directed graphs. We construct examples of such frames with arbitrarily many vectors.     

Perturbations of frames

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases. Similar results are also extended to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.