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1.
Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames and ( a discrete abelian group), relating the decay of the expansion of the elements of in terms of the elements of via a map . A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of , the relative measure of — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set in . Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.  相似文献   

2.
The R-dual sequences of a frame {f i } iI , introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ω j } jI to be an R-dual of a given frame {f i } iI . In particular we show that the R-duals {ω j } jI can be characterized in terms of frame properties of an associated sequence {n i } iI . We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.  相似文献   

3.
A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L(ℝ) such that {wgmn} (resp. ωψk,ℝ) is a normalized tight frame.  相似文献   

4.
Basic facts for Gabor frame {Eu(m)bTu(n)ag}m,n∈p on local field are investigated. Accurately, that the canonical dual of frame {Eu(m)bTu(n)ag}m,n∈p also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu(m)bTu(n)ag}m,n∈p to be a frame for L2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L2(K) are established. An example is finally given.  相似文献   

5.
In this paper, we present the conditions on dilation parameter {s j}j that ensure a discrete irregular wavelet system {s j n/2ψ(s j ·−bk)} j∈ℤ,k∈ℤ n to be a frame on L2(ℝn), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.  相似文献   

6.
This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.  相似文献   

7.
Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for 1-localized frames. We then specialize our results to Gabor multi-frames with generators in M 1(R d ), and Gabor molecules with envelopes in W(C, l 1). As a main tool in this work, we show there is a universal function g(x) so that, for every ε =s> 0, every Parseval frame {f i } i=1 M for an N-dimensional Hilbert space H N has a subset of fewer than (1+ε)N elements which is a frame for H N with lower frame bound g(ε/(2M/N − 1)). This work provides the first meaningful quantative notion of redundancy for a large class of infinite frames. In addition, the results give compelling new evidence in support of a general definition of redundancy given in [5].  相似文献   

8.
Recently people proved that every f∈C[0,1] can be uniformly approximated by polynomial sequences {Pn}, {Qn} such for any x∈[0,1] and n=1,2,… that {fx98-1}. For example, Xie and Zhou[2] showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous function. Actually they obtained a result as {fx98-2}, which essentially improved a conclusion in Gal and Szabados[1]. The present paper, by optimal procedure, improves this inequality to {fx98-3}, where ɛ is any positive real number.  相似文献   

9.
In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {A t  | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f t of G under which #{t∈ℕ | f t xA t } is infinite for a.e. xG/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov. Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999  相似文献   

10.
Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M) the elasticity of M. Define \overline{ρ}(x) = limn→∞ ρ(xn) to be the asymptotic elasticity of x. We determine some basic properties of the function \overline{ρ} and determine its image for certain block monoids.  相似文献   

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