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.     

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 construction of highly symmetric tight frames and complex polytopes

Many “highly symmetric” configurations of vectors in C^d${\mathbb{C}}^d$, such as the vertices of the platonic solids and the regular complex polytopes, are equal-norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group. For nonabelian groups there are uncountably many such tight frames up to unitary equivalence. The aim of this paper is to single out those orbits which are particularly nice, such as those which are the vertices of a complex polytope. This is done by defining a finite class of tight frames of n   vectors for C^d${\mathbb{C}}^d$ (n and d fixed) which we call the highly symmetric tight frames. We outline how these frames can be calculated from the representations of abstract groups using a computer algebra package. We give numerous examples, with a special emphasis on those obtained from the (Shephard–Todd) finite reflection groups. The interrelationships between these frames with complex polytopes, harmonic frames, equiangular tight frames, and Heisenberg frames (maximal sets of equiangular lines) are explored in detail.     

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.     

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.     

Robustness and surgery of frames

We characterize frames in Rⁿ that are robust to k erasures. The characterization is given in terms of the support of the null space of the synthesis operator of the frame. A necessary and sufficient condition is given for when an (r, k)-surgery on unit-norm tight frames in R² are possible. Also a generalization of a known characterization of the existence of tight frames with prescribed norms is given.     

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.     

Finite extensions of generalized Bessel sequences to generalized frames

The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.     

A unified characterization of reproducing systems generated by a finite family

This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L²(ℝⁿ). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.     

The number of harmonic frames of prime order

Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed between inequivalent harmonic frames and the orbits of a particular set. Secondarily, the symmetry group of prime order harmonic frames is shown to contain a subgroup consisting of a diagonal matrix as well as a permutation matrix, each of which is dependent on the particular harmonic frame in question.     

Estimates, decay properties, and computation of the dual function for Gabor frames

We present a simple proof of Ron and Shen's frame bounds estimates for Gabor frames. The proof is based on the Heil and Walnut's representation of the frame operator and shows that it can be decomposed into a continuous family of infinite matrices. The estimates then follow from a simple application of Gershgorin's theorem to each matrix. Next, we show that, if the window function has exponential decay, also the dual function has some exponential decay. Then, we describe a numerical method to compute the dual function and give an estimate of the error. Finally, we consider the spline of order 2; we investigate numerically the region of the time-frequency plane where it generates a frame and we compute the dual function for some values of the parameters.     

Finite-dimensional approximation of the inverse frame operator

A frame in a Hilbert space allows every element in to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of those coefficients and many other situations where frames occur, requires knowledge of the inverse frame operator. But usually it is hard to invert the frame operator if the underlying Hilbert space is infinite dimensional. In the present paper we introduce a method for approximation of the inverse frame operator using finite subsets of the frame. In particular this allows to approximate the frame coefficients (even inl ²) using finite-dimensional linear algebra. We show that the general method simplifies in the important cases of Weil-Heisenberg frames and wavelet frames.     

Dilation of Dual Frame Pairs in Hilbert C*-Modules

We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.     

Robust dimension reduction,fusion frames,and Grassmannian packings

We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first consider the linear minimum mean-squared error (LMMSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the mean-squared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equi-dimensional tight fusion frames. We call such fusion frames equi-distance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equi-distance tight fusion frames.     

Kernels of Gramian operators for frames in shift-invariant subspaces

For an invertible n×n matrix B and Φ a finite or countable subset of L²(Rⁿ), consider the collection X={?(·-Bk):?∈Φ,k∈Zⁿ} generating the closed subspace M of L²(Rⁿ). Our main objects of interest in this paper are the kernel of the associated Gramian G(.) and dual Gramian operator-valued functions. We show in particular that the orthogonal complement of M in L²(Rⁿ) can be generated by a Parseval frame obtained from a shift-invariant system having m generators where . Furthermore, this Parseval frame can be taken to be an orthonormal basis exactly when almost everywhere. Analogous results in terms of dim(Ker(G(.))) are also obtained concerning the existence of a collection of m sequences in the orthogonal complement of the range of analysis operator associated with the frame X whose shifts either form a Parseval frame or an orthonormal basis for that orthogonal complement.     

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.     

Subspace Weyl-Heisenberg frames

A Weyl-Heisenberg frame (WH frame) for L²(ℝ) allows every square integrable function on the line to be decomposed into the infinite sum of linear combination of translated and modulated versions of a fixed function. Some sufficient conditions for g ∈ L²(ℝ) to be a subspace Weyl-Heisenberg frame were given in a recent work [3] by Casazza and Christensen. Obviously every invariant subspace (under translation and modulation) is cyclic if it has a subspace WH frame. In the present article we prove that the cyclicity property is also sufficient for a subspace to admit a WH frame. We also investigate the dilation property for subspace Weyl-Heisenberg frames and show that every normalized tight subspace WH frame can be dilated to a normalized tight WH frame which is “maximal” In other words, every subspace WH frame is the compression of a WH frame which cannot be dilated anymore within the L²(ℝ) space. Communicated by Hans G. Feichtinger     

Tight Frames and Their Symmetries

The aim of this paper is to investigate the symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.     

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.