Density, Overcompleteness, and Localization of Frames. I. Theory

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.     

On the Duality Principle by Casazza, Kutyniok, and?Lammers

The R-dual sequences of a frame {f _i } _i∈I , 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 } _j∈I to be an R-dual of a given frame {f _i } _i∈I . In particular we show that the R-duals {ω _j } _j∈I can be characterized in terms of frame properties of an associated sequence {n _i } _i∈I . 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.     

Functional Gabor frame multipliers

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 {wg_mn} (resp. ωψ_k,ℝ) is a normalized tight frame.     

Basic Results of Gabor Frame on Local Fields

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.     

Irregular wavelet frames on<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript>n</Superscript>)

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.     

Measure functions for frames

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.     

A development on approximation by monotone sequences of polynomials

Recently people proved that every f∈C[0,1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} 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.     

Redundancy for localized frames

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 ℓ 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].     

Logarithm laws for flows on homogeneous spaces

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 x∈A _t } is infinite for a.e. x∈G/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     

Asymptotic Elasticity in Atomic Monoids

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) = lim_n→∞ 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.     

Riesz basis, Paley-Wiener class and tempered splines

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class.     

A unified characterization of reproducing systems generated by a finite family,II

By a “reproducing” method forH =L ²(ℝ ⁿ ) we mean the use of two countable families {e _α : α ∈A}, {f _α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e _α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e _α >:f _α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ ⁿ . Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.     

Gabor (super)frames with Hermite functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H _n . Let h = (H ₀, H ₁, . . . , H _n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices \({\Lambda \subseteq \mathbb{R} ^2}\) such that the Gabor system \({\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}\) is a frame for \({L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}\). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor frames.     

Product recurrent properties, disjointness and weak disjointness

Let F be a collection of subsets of ℝ₊ and (X, T) be a dynamical system; x ∈ X is F-recurrent if for each neighborhood U of x, {n ∈ ℝ₊: T ⁿ x ∈ U} ∈ F; x is F-product recurrent if (x, y) is recurrent for any F-recurrent point y in any dynamical system (Y, S). It is well known that x is {infinite}-product recurrent if and only if it is minimal and distal. In this paper it is proved that the closure of a {syndetic}-product recurrent point (i.e., weakly product recurrent point) has a dense minimal points; and a {piecewise syndetic}-product recurrent point is minimal. Results on product recurrence when the closure of an F-recurrent point has zero entropy are obtained.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

Amenable groups,stationary measures and partitions with independent iterates

Suppose a discrete amenable group G acts freely on a probability space (X, , μ) and {g _i } is any mixing sequence of group elements, that is μ(g _i ⁻¹ A ∩ B) → μ(A)μ(B) for all A, B ∈ . Then given any finite partition P and ε > 0 there is a subsequence {h _j } of {g _i } and a partition P′ differing from P on a set of measure less than ε such that the partitions {gP: g ∈ IP′{h _j }} are jointly independent, where IP′{h _j } denotes the set

On one extremal problem for numerical series

Let Γ be the set of all permutations of the natural series and let α = {α _j} _j∈ℕ, ν = {ν_j} _j∈ℕ, and η = {η_j} _j∈ℕ be nonnegative number sequences for which

Localization of Frames,Banach Frames,and the Invertibility of the Frame Operator

We introduce a new concept to describe the localization of frames. In our main result we show that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about non-uniform sampling in shift-invariant spaces.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.     

Some results about operators in nested Hilbert spaces

With the use of interpolation methods we obtain some results about the domain of an operator acting on the nested Hilbert space {ℋ_f}_f∈∑ generated by a self-adjoint operatorA and some estimates of the norms of its representatives. Some consequences in the particular case of the scale of Hilbert spaces are discussed.