The symmetry group of a finite frame

We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of Vale and Waldron (2005) [12] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames.     

广义框架和广义Riesz基的摄动

引入并研究了Banach空间X中的Bessel集、广义框架与广义Riesz基．对X中的任一Bessel集{gm}m∈M，定义有界线性算子T：L^2（P）→X^＊，利用算子丁，给出了Bessel集与广义框架的等价刻画．同时讨论了广义框架和广义Riesz基的摄动．     

Optimal dual frames for erasures

For any given frame (for the purpose of encoding) in a finite dimensional Hilbert space, we investigate its dual frames that are optimal for erasures (for the purpose of decoding). We show that in general the canonical dual is not necessarily optimal. Moreover, optimal dual frames are not necessarily unique. We present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem. As an application, we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight.     

Banach空间中的X_d框架与Reisz基

本文引入并研究了Banach空间中的X_d框架,X_d Bessel列,紧X_d框架,独立X_d框架和X_d Riesz基等概念,给出了X_d框架和独立X_d框架的算子等价刻画,Banach空间X中存在X_d框架或X_d Riesz基的充要条件以及X_d框架的对偶框架存在的充要条件,讨论了Banach空间的基和X_d框架,X_d Riesz基之间的关系．     

广义Frame与广义Frame的商

本文将frame、frame同态、商frame与核的概念在范畴意义下作推广,并且证明Frame范畴是广义Frame范畴的反射子范畴.进而讨论了一个广义frame A的商与A上核函子之间的关系.特别地,我们证明了A上全部核函子所构成的范畴N(A)是一个广义frame.     

$W^\ast$-模上的标准广义框架的和

In this paper, the sum of standard generalized flames of Hilbert W^＊-module is studied intensively by using operator-theoretic-methods, and some conditions are given to assure that the sum of two or more standard generalized frames is a standard generalized frame.     

广义框架和框架算子

本文研究了可分的Hilbert空间H中的广义框架,应用算子论方法给出了广义框架是H中紧广义框架,对偶广义框架,独立广义框架的充要条件:证明了有关广义框架算子的一些结果。     

Generation of finite tight frames by Householder transformations

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ?ⁿ or ?ⁿ only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ?ⁿ or ?ⁿ, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?     

基于Spectral Tetris算法的框架的最佳稀疏性

当信号维数较大时,使用稀疏框架分解信号就能减少大量的加法和乘法运算,所以,研究稀疏框架很有意义.本文介绍有限框架的稀疏性,并研究基于Spectral Tetris算法构造的框架的稀疏性.首先,给出基于Spectral Tetris算法的框架的最佳稀疏性;其次,得到基于Spectral Tetris算法的可剖分紧框架的最佳稀疏性.     

正交射影在广义框架理论中的应用

本文研究了可分的Hilbert空间H中的广义框架,运用算子理论方法,研究了可分的 Hilbert空间H中广义框架的性质,给出了广义框架的对偶广义框架的一些刻画,并且证明了两个广义框架是强非交的一个充分必要条件．     

Hilbert空间H中带符号广义框架的一个刻画

本文研究了可分的Hilbert空间H中带符号广义框架,利用算子理论方法,给出了H中一族向量{hm}m∈M是一个带符号广义框架当且仅当带符号广义框架的框架算子的正部S 和负部S-是有界线性算子,讨论了H中带符号广义框架的框架算子S的可逆性,并且得到了H中每个向量f关于带符号广义框架{hm}m∈M和其对偶带符号广义框架{~hm}m∈M的表示式.     

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.     

Gabor frames, unimodularity, and window decay

We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties of g are inherited by its minimal dual γ⁰ and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying (i.e., decay like |g(t)|≤Ce^{−α|t| 1/α} for some 1/2≤α<1) window functions. Particularly, we find that g and γ⁰ have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e., its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on results from the theory of entire functions, and on the theory of polynomial matrices.     

Frames of Periodic Shift-Invariant Spaces

This note studies Bessel sequences and frames of shift-invariant spaces generated by a countable set of periodic functions. We give characterizations under which the set of translations of the countable set is a Bessel sequence or a frame in terms of spectral decompositions of some self-adjoint operators.     

Generalized tight p-frames and spectral bounds for Laplace-like operators

We prove sharp upper bounds for sums of eigenvalues (and other spectral functionals) of Laplace-like operators, including bi-Laplacians and fractional Laplacians. We show that among linear images of a highly symmetric domain, our spectral functionals are maximal on the original domain. We exploit the symmetries of the domain, and the operator, avoiding the necessity of finding good test functions for variational problems. This is especially important for fractional Laplacians, since exact solutions are not even known on intervals, making it hard to find good test functions.To achieve our goals we generalize tight p-fusion frames, to extract the best possible geometric results for domains with isometry groups admitting tight p-frames. Any such group generates a tight p-fusion frame via conjugations of a fixed projection matrix. We show that generalized tight p-frames can also be obtained by conjugations of an arbitrary rectangular matrix, with the frame constant depending on the singular values of the matrix.     

On the Orthogonality of Frames and the Density and Connectivity of Wavelet Frames

We examine some recent results of Bownik on density and connectivity of the wavelet frames. We use orthogonality (strong disjointness) properties of frame and Bessel sequences, and also properties of Bessel multipliers (operators that map wavelet Bessel functions to wavelet Bessel functions). In addition we obtain an asymptotically tight approximation result for wavelet frames.     

Tight wavelet frames for subdivision

In this paper we construct multivariate tight wavelet frame decompositions for scalar and vector subdivision schemes with nonnegative masks. The constructed frame generators have one vanishing moment and are obtained by factorizing certain positive semi-definite matrices. The construction is local and allows us to obtain framelets even in the vicinity of irregular vertices. Constructing tight frames, instead of wavelet bases, we avoid extra computations of the dual masks. In addition, the frame decomposition algorithm is stable as the discrete frame transform is an isometry on _?2${?}_2$, if the data are properly normalized.     

Bessel框架及其摄动

本文进一步研究了Bessel框架的性质,给出其稳定性和摄动的几个有意义的结果。     

Random Tight Frames

We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere approximately form a finite unit norm tight frame (FUNTF). In the present paper, we develop probabilistic versions of tight frames and FUNTFs to significantly weaken the requirements on the random choice of points to obtain an approximate finite tight frame. Namely, points can be chosen from any probabilistic tight frame, they do not have to be identically distributed, nor have unit norm. We also observe that classes of random matrices used in compressed sensing are induced by probabilistic tight frames.