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Peter G. Casazza

Publications and source records attributed to Peter G. Casazza.

At least 73 records · Page 4Linked to original sources
arXiv2009

Fusion Frames: Existence and Construction

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods -- the spatial complement and the Naimark complement -- for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given $M, N, m \in \NN$ and $\{λ_j\}_{j=1}^M$, does there exist a fusion frame in $\RR^M$ with $N$ subspaces of dimension $m$ for which $\{λ_j\}_{j=1}^M$ are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters $M, N, m \in \NN$ and $\{λ_j\}_{j=1}^M$. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.

math.FA↗
arXiv2009

Infinite dimensional restricted invertibility

The 1987 Bourgain-Tzafriri Restricted Invertibility Theorem is one of the most celebrated theorems in analysis. At the time of their work, the authors raised the question of a possible infinite dimensional version of the theorem. In this paper, we will give a quite general definition of restricted invertibility for operators on infinite dimensional Hilbert spaces based on the notion of "density" from frame theory. We then prove that localized Bessel systems have large subsets which are Riesz basic sequences. As a consequence, we prove the strongest possible form of the infinite dimensional restricted invertibility theorem for $\ell_1$-localized operators and for Gabor frames with generating function in the Feichtinger Algebra. For our calculations, we introduce a new notion of "density" which has serious advantages over the standard form because it is independent of index maps - and hence has much broader application. We then show that in the setting of the restricted invertibility theorem, this new density becomes equivalent to the standard density.

math.FA↗
arXiv2007

A Decomposition Theorem for frames and the Feichtinger Conjecture

In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in $C^{*}$-Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is $ω$-independent for $\ell_2$-sequences.

math.FA↗
arXiv2007

Projections and the Kadison-Singer Problem

We prove some new equivalences of the paving conjecture and obtain some estimates on the paving constants. In addition we give a new family of counterexamples to one of the Akemann-Anderson conjectures.

math.FA↗
arXiv2006

Fusion Frames and Distributed Processing

Let $\{W_i\}_{i\in I}$ be a (redundant) sequence of subspaces each being endowed with a weight $v_i$, and let $\mathcal{H}$ be the closed linear span of the $W_i$'s, a composite Hilbert space. Provided that $\{(W_i,v_i)\}_{i \in I}$ satisfies a certain property which controls the weighted overlaps of the subspaces, it is called a {\em fusion frame}. These systems contain conventional frames as a special case, however they go far ``beyond frame theory''. In case each subspace $W_i$ is equipped with a frame system $\{f_{ij}\}_{j \in J_i}$ by which it is spanned, we refer to $\{(W_i,v_i,\{f_{ij}\}_{j \in J_i})\}_{i \in I}$ as a {\em fusion frame system}. In this paper, we describe a weighted and distributed processing procedure that fuse together information in all subspaces $W_i$ of a fusion frame system to obtain the global information in $\mathcal{H}$. The weighted and distributed processing technique described in fusion frames is not only a natural fit in distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. We further provide an extensive study of the robustness of fusion frame systems.

math.FA↗
arXiv2006

The Kadison-Singer Problem in Mathematics and Engineering

We will show that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.

math.FA↗
arXiv2003

Frames of subspaces

One approach to ease the construction of frames is to first construct local components and then build a global frame from these. In this paper we will show that the study of the relation between a frame and its local components leads to the definition of a frame of subspaces. We introduce this new notion and prove that it provides us with the link we need. It will also turn out that frames of subspaces behave as a generalization of frames. In particular, we can define an analysis, a synthesis and a frame operator for a frame of subspaces, which even yield a reconstruction formula. Also concepts such as completeness, minimality, and exactness are introduced and investigated. We further study several constructions of frames of subspaces, and also of frames and Riesz frames using the theory of frames of subspaces. An important special case are harmonic frames of subspaces which generalize harmonic frames. We show that wavelet subspaces coming from multiresolution analysis belong to this class.

math.FA↗
arXiv2000

Perturbations of Weyl-Heisenberg frames

We develop a usable perturbation theory for Weyl-Heisenberg frames. In particular, we prove that if $(E_{mb}T_{na}g)_{m,n\inmathbb Z}$ is a WH-frame and $h$ is a function which is close to $g$ in the Wiener Amalgam space norm, then $(E_{mb}T_{na}h)_{m,n\in \mathbb Z}$ is also a WH-frame.

math.FA↗
arXiv1999

Bracket products for Weyl-Heisenberg frames

We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions $(g_{n})$ into a sequence $(e_{n})$ with the property that $(E_{mb}e_{n})_{m,n\in \Bbb Z}$ is orthonormal in $L^{2}(\Bbb R)$. Armed with this inner product, we obtain several results concerning Weyl-Heisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions $g\in L^{2}(\Bbb R)$ and $ab=1$ so that the family $(E_{mb}T_{na}g)$ is complete in $L^{2}(\Bbb R)$. One consequence of this is that for functions $g$ supported on a half-line $[α,\infty)$ (in particular, for compactly supported $g$), $(g,1,1)$ is complete if and only if $\text{sup}_{0\le t< a}|g(t-n)|\not= 0$ a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any $g\in L^{2}(\Bbb R)$, $A\le \sum_{n} |g(t-na)|^{2}\le B$ is equivalent to $(E_{m/a}g)$ being a Riesz basic sequence.

math.FA↗
arXiv1999

Analyzing the Weyl-Heisenberg Frame Identity

In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg Frame Identity. WH-Frame Identity: If $g\in W(L^{\infty},L^{1})$, then for all continuous, compactly supported functions f we have: \[\sum_{m,n}| |^{2} = \frac{1}{b}\sum_{k}\int_{\Bbb R}\bar{f(t)}f(t-k/b)\sum_{n} g(t-na)\bar{g(t-na-k/b)} dt.\] It has been folklore that the identity will not hold universally. We make a detailed study of the WH-Frame Identity and show: (1) The identity does not require any assumptions on ab (such as the requirement that $ab\le 1$ to have a frame); (2) As stated above, the identity holds for all $f\in L^{2}(\Bbb R)$; (3) The identity holds for all bounded, compactly supported functions if and only if $g\in L^{2}(\Bbb R)$; (4) The identity holds for all compactly supported functions if and only if $\sum_{n}|g(x-na)|^{2}\le B$ a.e.; Moreover, in (2)-(4) above, the series on the right converges unconditionally; (5) In general, there are WH-frames and functions $f\in L^{2}(\Bbb R)$ so that the series on the right does not converge (even symmetrically). We give necessary and sufficient conditions for it to converge symmetrically; (6) There are WH-frames for which the series on the right always converges symmetrically to give the WH-Frame Identity, but there are functions for which the series does not converge and we classify when the series converges for all functions $f\in Ł$; (7) There are WH-frames for which the series always converges, but it does not converge unconditionally for some functions, and we classify when we have unconditional convergence for all functions f; and (8) We show that the series converges unconditionally for all $f\in L^{2}(\Bbb R)$ if g satisfies the CC-condition.

math.FA↗
arXiv1999

The Art of Frame Theory

Thye theory of frames for a Hilbert space plays a fundamental role in signal processing, image processing, data compression, sampling theory and much more, as well as being a fruitful area of research in abstract mathematics. In this ``tutorial'' on abstract frame theory, we will try to point out the major directions of research in abstract frame theory and give some sample techniques from each of the areas. We will also bring out some of the important open questions, discuss some of the limitations of the existing theory, and point to some new directions for research.

math.FA↗
arXiv1999

Weyl-Heisenberg Frames, Translation Invariant Systems and the Walnut Representation

We present a comprehensive analysis of the convergence properties of the frame operators of Weyl-Heisenberg systems and shift-invariant systems, and relate these to the convergence of the Walnut representation. We give a deep analysis of necessary conditions and sufficient conditions for convergence of the frame operator. We show that symmetric, norm and unconditional convergence of the Walnut series are all different, but that weak and norm convergence are the same, while there are WH-systems for which the Walnut representation has none of these convergence properties. We make a detailed study of the CC-Condition (a sufficient condition for WH-systems to have finite upper frame bounds), and show that (for ab rational) a uniform version of this passes to the Wexler-Raz dual. We also show that a condition of Tolimieri and Orr implies the uniform CC-Condition. We obtain stronger results in the case when (g,a,b) is a WH-system and ab is rational. For example, if ab is rational, then the CC-Condition becomes equivalent to the unconditional convergence of the Walnut representation - even in a more general setting. Many of the results are generalized to shift-invariant systems. We give classifications for numerous important classes of WH-systems including: (1) The WH-systems for which the frame operator extends to a bounded operator on L^p(R), 1\leq p; (2) The WH-systems for which the frame operator extends to a bounded operator on the Wiener amalgam space; (3) The families of frames which have the same frame operator.

math.FA↗
arXiv1999

The solution to the Maurey extension problem for Banach spaces with the Gordon Lewis property and related structures

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then $B(\ell_{\infty},X^*)=Π_2(\ell_{\infty},X^*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach lattice) or if X is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension property for these classes of spaces. The paper also contains a detailed study of the property of extending operators with values in $\ell_p$-spaces, $1\le p<\infty$.

math.FA↗
arXiv1998

Classifying Tight Weyl-Heisenberg Frames

A Weyl-Heisenberg frame for L^2(R) is a frame consisting of translates and modulates of a fixed function. In this paper we give necessary and sufficient conditions for this family to form a tight WH-frame. This allows us to write down explicitly all functions g for which all translates and modulates of g form an orthonormal basis for L^2(R). There are a number of consequences of this classification, including a simple direct classification of the alternate dual frames to a WH-frame (A result originally due to Janssen).

math.FA↗
arXiv1998

Embeddings of Banach Spaces into Banach Lattices and the Gordon-Lewis Property

In this paper we first show that if $X$ is a Banach space and $α$ is a left invariant crossnorm on $\ell_\infty\otimes X$, then there is a Banach lattice $L$ and an isometric embedding $J$ of $X$ into $L$, so that $I\otimes J$ becomes an isometry of $\ell_\infty\otimes_αX$ onto $\ell_\infty\otimes_m J(X)$. Here $I$ denotes the identity operator on $\ell_\infty$ and $\ell_\infty\otimes_m J(X)$ the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to characterize the Gordon-Lewis property $\GL$ in terms of embeddings into Banach lattices. Also other structures related to the $\GL$ are investigated.

math.FA↗
arXiv1998

Frames of translates

We give necessary and sufficient conditions for a subfamily of regularly spaced translates of a function to form a frame (resp. a Riesz basis) for its span. One consequence is that ifthetranslates are taken only from a subset of the natural numbers, then this family is a frame if and only if it is a Riesz basis. We also consider arbitrary sequences of translates and show that for sparse sets, having an upper frame bound is equivalent to the family being a frame sequence. Finally, we use the fractional Hausdorff dimension to identify classes of exact frame sequences.

math.FA↗