DOCSLIB.ORG

A Brief Introduction to Hilbert Space Frame Theory and Its Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Brief Introduction to Hilbert Space Frame Theory and Its Applications A BRIEF INTRODUCTION TO HILBERT SPACE FRAME THEORY AND ITS APPLICATIONS PETER G. CASAZZA AND RICHARD G. LYNCH Abstract. This is a short introduction to Hilbert space frame theory and its applications for those outside the area who want to enter the subject. We will emphasize finite frame theory since it is the easiest way to get into the subject. Contents 1. Reading List 2 2. The Basics of Hilbert Space Theory 2 3. The Basics of Operator Theory 6 4. Hilbert Space Frames 14 4.1. Frame Operators 15 4.2. Dual Frames 20 4.3. Redundancy 21 4.4. Minimal Moments 21 4.5. Orthogonal Projections and Naimark’s Theorem 22 4.6. Frame Representations 23 5. Constants Related to Frames 23 6. Constructing Finite Frames 24 6.1. Finding Parseval Frames 25 6.2. Adding vectors to a frame to make it tight 25 6.3. Majorization 25 6.4. Spectral Tetris 26 7. Gramian Operators 29 8. Fusion Frames 30 9. Infinite Dimensional Hilbert Spaces 32 9.1. Hilbert Spaces of Sequences 32 9.2. Hilbert Spaces of Functions 32 10. Major Open Problems in Frame Theory 33 10.1. Equiangular Frames 33 arXiv:1509.07347v2 [math.FA] 14 Feb 2016 10.2. The Scaling Problem 35 10.3. Sparse Orthonormal Bases for Subspaces 35 10.4. The Paulsen Problem 36 10.5. Concrete Construction of RIP Matrices 36 10.6. An Algorithm for the Feichtinger Conjecture 37 10.7. Classifying Gabor Frames 37 10.8. Phase Retrieval 38 References 39 The authors were supported by NSF DMS 1307685; and NSF ATD 1042701 and 1321779; AFOSR DGE51: FA9550-11-1- 0245. 1 2 P.G. CASAZZA AND R.G. LYNCH 1. Reading List For a more complete treatment of frame theory we recommend the books of Han, Kornelson, Larson, and Weber [59], Christensen [41], the book of Casazza and Kutyniok [32], the tutorials of Casazza [23, 24] and the memoir of Han and Larson [60]. For a complete treatment of frame theory in time-frequency analysis we recommend the book of Gr¨ochenig [55]. For an introduction to frame theory and filter banks plus applications to engineering we recommend Kovaˇcevi´cand Chebira [64]. Also, a wealth of information can be found at the Frame Research Center’s website [53]. 2. The Basics of Hilbert Space Theory Given a positive integer N, we denote by HN the real or complex Hilbert space of dimension N. This is either RN or CN with the inner product given by N x, y = a b h i i i i=1 X for x = (a ,a , ,a ) and y = (b ,b , ,b ) and the norm of a vector x is 1 2 ··· N 1 2 ··· N x 2 = x, x . k k h i For x, y HN , x y is the distance from the vector x to the vector y. For future reference, note that in the real∈ case, k − k x y 2 = x y, x y k − k h − − i = x, x x, y y, x + y,y h i − h i − h i h i = x 2 2 x, y + y 2 k k − h i k k and in the complex case we have x, y + y, x = x, y + x, y =2Re x, y , h i h i h i h i h i where Rec denotes the real part of the complex number c. We will concentrate on finite dimensional Hilbert spaces since it is the easiest way to get started on the subject of frames. Most of these results hold for infinite dimensional Hilbert spaces and at the end we will look at the infinite dimensional case. The next lemma contains a standard trick for calculations. Lemma 2.1. If x HN and x, y =0 for all y HN then x =0. ∈ h i ∈ Proof. Letting y = x we have 0= x, y = x, x = x 2, h i h i k k and so x = 0. Definition 2.2. A set of vectors e M in HN is called: { i}i=1 (1) linearly independent if for any scalars a M , { i}i=1 M a e =0 a =0, for all i =1, 2, , M. i i ⇒ i ··· i=1 X Note that this requires ei = 0 for all i =1, 2, ,M. (2) complete (or a spanning6 set) if ··· span e M = HN . { i}i=1 (3) orthogonal if for all i = j, ei,ej =0. (4) orthornormal if it is orthogonal6 h i and unit norm. (5) an orthonormal basis if it is complete and orthonormal. The following is immediate from the definitions. FRAMES AND THEIR APPLICATIONS 3 Proposition 2.3. If e N is an orthonormal basis for HN , then for every x H we have { i}i=1 ∈ N x = x, e e . h ii i i=1 X From the previous proposition, we can immediately deduce an essential identity called Parseval’s Identity. N HN HN Proposition 2.4 (Parseval’s Identity). If ei i=1 is an orthonormal basis for , then for every x , we have { } ∈ N x 2 = x, e 2. k k |h ii| i=1 X Some more basic identities and inequalities for Hilbert space that are frequently used are contained in the next proposition. Proposition 2.5. Let x, y HN . ∈ (1) Cauchy-Schwarz Inequality: x, y x y , |h i|≤k kk k with equality if and only if x = cy for some constant c. (2) Triangle Inequality: x + y x + y . k k≤k k k k (3) Polarization Identity: Assuming HN is real, 1 x, y = x + y 2 x y 2 . h i 4 k k −k − k If HN is complex, then 1 x, y = x + y 2 x y 2 + i x + iy 2 i x iy 2 . h i 4 k k −k − k k k − k − k (4) Pythagorean Theorem: Given pairwise orthogonal vectors x M , { i}i=1 M 2 M 2 xi = xi . k k i=1 i=1 X X Proof. (1) The inequality is trivial if y = 0. If y = 0, we may assume y = 1 by dividing through the inequality with y . Now we compute: 6 k k k k 0 < x x, y y 2 k − h i k = x 2 2 x, y x, y + x, y 2 y 2 k k − h ih i |h i| k k = x 2 x, y 2 k k − |h i| = x 2 y 2 x, y 2. k k k k − |h i| Note that the strict inequality would be equality if x = cy. (2) Applying (1) to obtain the second inequality: x + y 2 = x 2 +2Re x, y + y 2 k k k k h i k k x 2 +2 x, y + y 2 ≤k k |h i| k k x 2 +2 x y + y 2 ≤k k k kk k k k = ( x + y )2. k k k k (3) We compute assuming HN is a real Hilbert space: x + y 2 x y 2 = x 2 +2 x, y + y 2 ( x 2 2 x, y + y 2) k k −k − k k k h i k k − k k − h i k k =4 x, y . h i The proof in the complex case is similar. 4 P.G. CASAZZA AND R.G. LYNCH (4) Since x , x = 0 for all i = j, we have h i j i 6 M 2 M M xi = xi, xj * + i=1 i=1 j=1 X X X M = x , x h i j i i,j=1 X M = x , x h i ii i=1 X M = x 2. k ik i=1 X We now look at subspaces of the Hilbert space. Definition 2.6. Let W, V be subspaces of HN . (1) A vector x HN is orthogonal to a subspace W , denoted x W if ∈ ⊥ x, y = 0 for all y W. h i ∈ The orthogonal complement of W is W ⊥ = x H : x W . { ∈ ⊥ } (2) The subspaces W, V are orthogonal subspaces, denoted W V if W V ⊥, that is, ⊥ ⊂ x, y = 0 for all x W, y V. h i ∈ ∈ A simple calculation shows that W ⊥ is always closed and so if W is closed then W ⊥⊥ = W . Fundamental to Hilbert space theory are orthogonal projections as defined next. Definition 2.7. An operator P : HN HN is called a projection if P 2 = P . It is an orthogonal projection if P is also self-adjoint (See definition→ 3.3). For any subspace W HN , there is an orthogonal projection of H onto W called the nearest point projection. One way to⊂ define it is to pick any orthonormal basis e K for W and define { i}i=1 K P x = x, e e . h ii i i=1 X Note that for all j =1, 2, ,K we have ··· K K Px,e = x, e e ,e = x, e e ,e = x, e . h j i h ii i j h iih i ji h j i i=1 i=1 X X We need to check that this operator is well defined, that is, we must show it is independent of the choice of basis. Lemma 2.8. If e K and g K are orthonormal bases for W , then { i}i=1 { i}i=1 K K x, e e = x, g g . h ii i h ii i i=1 i=1 X X Proof. We compute: K K K x, e e = x, e ,g g e h ii i h i j i j i i=1 i=1 * j=1 + X X X K = e ,g x, g e h i jih j i i i,j=1 X FRAMES AND THEIR APPLICATIONS 5 K K = x, g g ,e e h j i j i i i=1 *j=1 + X X K = x, g g .
Load more

Recommended publications
  • Sums of Hilbert Space Frames 1
    SUMS OF HILBERT SPACE FRAMES SOFIAN OBEIDAT, SALTI SAMARAH, PETER G. CASAZZA AND JANET C. TREMAIN Abstract. We give simple necessary and sufficient conditions on Bessel sequences {fi} and {gi} and operators L1,L2 on a Hilbert space H so that {L1fi + L2gi} is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames. 1. Introduction Frames for Hilbert spaces were introduced by Duffin and Schaeffer [8] as a part of their research in non-harmonic Fourier series. Their work on frames was somewhat forgotten until 1986 when Daubechies, Grossmann and Meyer [13] brought it all back to life during their fundamental work on wavelets. Today, frame theory plays an important role not just in signal processing, but also in dozens of applied areas (See [1, 2]). Holub [12] showed that if {xn} is any normalized basis for a Hilbert space H and {fn} is the associated dual basis of coefficient functionals, then the sequence {xn + fn} is again a basis for H. In this paper we study cases in which new frames can be obtained from old ones. Throughout H denotes a separable Hilbert space and HN is the Nth dimensional Hilbert space. A frame for H is a family of vectors fi ∈ H, for i ∈ I for which there exist constants A, B > 0 satisfying: 2 X 2 2 (1.1) Akfk ≤ |hf, fii| ≤ Bkfk i∈I for all f ∈ H. A and B are called the lower and upper frame bounds respectively. If A = B, this is called an A-tight frame.
    [Show full text]
  • Designing Structured Tight Frames Via an Alternating Projection Method Joel A
    1 Designing Structured Tight Frames via an Alternating Projection Method Joel A. Tropp, Student Member, IEEE, Inderjit S. Dhillon, Member, IEEE, Robert W. Heath Jr., Member, IEEE and Thomas Strohmer Abstract— Tight frames, also known as general Welch-Bound- tight frame to a given ensemble of structured vectors; then Equality sequences, generalize orthonormal systems. Numer- it finds the ensemble of structured vectors nearest to the ous applications—including communications, coding and sparse tight frame; and it repeats the process ad infinitum. This approximation—require finite-dimensional tight frames that pos- sess additional structural properties. This paper proposes a technique is analogous to the method of projection on convex versatile alternating projection method that is flexible enough to sets (POCS) [4], [5], except that the class of tight frames is solve a huge class of inverse eigenvalue problems, which includes non-convex, which complicates the analysis significantly. Nev- the frame design problem. To apply this method, one only needs ertheless, our alternating projection algorithm affords simple to solve a matrix nearness problem that arises naturally from implementations, and it provides a quick route to solve difficult the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. frame design problems. This article offers extensive numerical Alternating projection is likely to succeed even when algebraic evidence that our method succeeds for several important cases. constructions are unavailable. We argue that similar techniques apply to a huge class of To demonstrate that alternating projection is an effective tool inverse eigenvalue problems. for frame design, the article studies some important structural There is a major conceptual difference between the use properties in detail.
    [Show full text]
  • FRAME WAVELET SETS in R §1. Introduction a Collection of Elements
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 7, Pages 2045{2055 S 0002-9939(00)05873-1 Article electronically published on December 28, 2000 FRAME WAVELET SETS IN R X. DAI, Y. DIAO, AND Q. GU (Communicated by David R. Larson) Abstract. In this paper, we try to answer an open question raised by Han and Larson, which asks about the characterization of frame wavelet sets. We completely characterize tight frame wavelet sets. We also obtain some nec- essary conditions and some sufficient conditions for a set E to be a (general) frame wavelet set. Some results are extended to frame wavelet functions that are not defined by frame wavelet set. Several examples are presented and compared with some known results in the literature. x1. Introduction A collection of elements fxj : j 2Jgin a Hilbert space H is called a frame if there exist constants A and B,0<A≤ B<1, such that X 2 2 2 (1) Akfk ≤ jhf;xj ij ≤ Bkfk j2J for all f 2H. The supremum of all such numbers A and the infimum of all such numbers B are called the frame bounds of the frame and denoted by A0 and B0 respectively. The frame is called a tight frame when A0 = B0 andiscalleda normalized tight frame when A0 = B0 = 1. Any orthonormal basis in a Hilbert space is a normalized tight frame. The concept of frame first appeared in the late 40's and early 50's (see [6], [13] and [12]). The development and study of wavelet theory during the last decade also brought new ideas and attention to frames because of their close connections.
    [Show full text]
  • The Contourlet Transform: an Efficient Directional Multiresolution Image Representation Minh N
    IEEE TRANSACTIONS ON IMAGE PROCESSING 1 The Contourlet Transform: An Efficient Directional Multiresolution Image Representation Minh N. Do, Member, IEEE, and Martin Vetterli, Fellow, IEEE Abstract— The limitations of commonly used separable ex- representation has to be obtained by structured transforms and tensions of one-dimensional transforms, such as the Fourier fast algorithms. and wavelet transforms, in capturing the geometry of image For one-dimensional piecewise smooth signals, like scan- edges are well known. In this paper, we pursue a “true” two- dimensional transform that can capture the intrinsic geometrical lines of an image, wavelets have been established as the right structure that is key in visual information. The main challenge tool, because they provide an optimal representation for these in exploring geometry in images comes from the discrete nature signals in a certain sense [1], [2]. In addition, the wavelet of the data. Thus, unlike other approaches, such as curvelets, representation is amenable to efficient algorithms; in particular that first develop a transform in the continuous domain and it leads to fast transforms and convenient tree data structures. then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence These are the key reasons for the success of wavelets in to an expansion in the continuous domain. Specifically, we many signal processing and communication applications; for construct a discrete-domain multiresolution and multidirection example, the wavelet transform was adopted as the transform expansion using non-separable filter banks, in much the same way for the new image-compression standard, JPEG-2000 [3].
    [Show full text]
  • The Canonical Dual Frame of a Wavelet Frame
    Applied and Computational Harmonic Analysis 12, 269–285 (2002) doi:10.1006/acha.2002.0381 The Canonical Dual Frame of a Wavelet Frame Ingrid Daubechies 1 PACM, Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544 E-mail: [email protected] and Bin Han 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada E-mail: [email protected] Communicated by Charles K. Chui Received July 11, 2000; revised December 20, 2001 In this paper we show that there exist wavelet frames that have nice dual wavelet frames, but for which the canonical dual frame does not consist of wavelets, i.e., cannot be generated by the translates and dilates of a single function. 2002 Elsevier Science (USA) Key Words: wavelet frame; the canonical dual frame; dual wavelet frame. 1. INTRODUCTION We start by recalling some notations and definitions. Let H be a Hilbert space. A set of elements {hk}k∈Z in H is said to be a frame (see [5]) in H if there exist two positive constants A and B such that 2 2 2 Af ≤ |f,hk| ≤ Bf , ∀f ∈ H, (1.1) k∈Z where A and B are called the lower frame bound and upper frame bound, respectively. In particular, when A = B = 1, we say that {hk}k∈Z is a (normalized) tight frame in H.The 1 Research was partially supported by NSF Grants DMS-9872890, DMS-9706753 and AFOSR Grant F49620- 98-1-0044. Web: http://www.princeton.edu/~icd. 2 Research was supported by NSERC Canada under Grant G121210654 and by Alberta Innovation and Science REE under Grant G227120136.
    [Show full text]
  • Quantization in Signal Processing with Frame Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Vanderbilt Electronic Thesis and Dissertation Archive QUANTIZATION IN SIGNAL PROCESSING WITH FRAME THEORY By JIAYI JIANG Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics May, 2016 Nashville, Tennessee Approved: Alexander Powell Akram Aldroubi Doug Hardin Brett Byram ©Copyright by Jiayi Jiang, 2016. All Rights Reserved ii ACKNOWLEDGEMENTS Firstly, I would like to express my sincere gratitude to my advisor, Alexander.M.Powell. He taught me too many thing in these five years which are not only on mathematics research but also on how to be a polite and organised person. Thanks a lot for his encouragement, patience and professional advices. I really appreciate for his supporting and helping when I decide to work in industry instead of academic. All of these mean a lot to me! My time in Vanderbilt has been a lot of fun. Thanks for the faculty of the math depart- ment. Thanks for all my teachers, Akram Aldroubi, Doug Hardin, Mark Ellingham, Mike Mihalik, Edward Saff. Each Professors in these list taught me at least one course which I will be benefited from them in my future carer. I really enjoy the different way of teaching from these professors. Thanks for my Vandy friends, Corey Jones, Yunxiang Ren, Sui tang and Zhengwei Liu. You guys give me too many helps and joys. I have to say thanks to my wife Suyang Zhao.
    [Show full text]
  • Tight Wavelet Frame Construction and Its Application for Image Processing
    Tight Wavelet Frame Construction and its application for Image Processing by Kyunglim Nam (Under the direction of Ming-Jun Lai) Abstract As a generalized wavelet function, a wavelet frame gives more rooms for different construc- tion methods. In this dissertation, first we study two constructive methods for the locally supported tight wavelet frame for any given refinable function whose Laurent polynomial satisfies the QMF or the sub-QMF conditions in Rd. Those methods were introduced by Lai and St¨ockler. However, to apply the constructive method under the sub-QMF condition we need to factorize a nonnegative Laurent polynomial in the multivariate setting into an expression of a finite square sums of Laurent polynomials. We find an explicit finite square sum of Laurent polynomials that expresses the nonnegative Laurent polynomial associated with a 3-direction or 4-direction box spline for various degrees and smoothness. To facilitate the description of the construction of box spline tight wavelet frames, we start with B-spline tight wavelet frame construction. For B-splines we find the sum of squares form by using Fej´er-Rieszfactorization theorem and construct tight wavelet frames. We also use the tensor product of B-splines to construct locally supported bivariate tight wavelet frames. Then we explain how to construct box spline tight wavelet frames using Lai and St¨ockler’s method. In the second part of dissertation, we apply some of our box spline tight wavelet frames for edge detection and image de-noising. We present a lot of images to compare favorably with other edge detection methods including orthonormal wavelet methods and six engineering methods from MATLAB Image Processing Toolbox.
    [Show full text]
  • The Complex Shearlet Transform and Applications to Image Quality Assessment
    Technische Universität Berlin Fachgruppe Angewandte Funktionalanalysis The Complex Shearlet Transform and Applications to Image Quality Assessment Master Thesis by Rafael Reisenhofer 18. Juli 2014 – 17. Oktober 2014 Primary reviewer: Prof. Dr. Gitta Kutyniok Secondary reviewer: Dr. Wang-Q Lim Supervisor: Prof. Dr. Gitta Kutyniok Rafael Reisenhofer Beusselstraße 64 10553 Berlin Hiermit erkläre ich, dass ich die vorliegende Arbeit selbstständig und eigenhändig sowie ohne unerlaubte fremde Hilfe und ausschließlich unter Verwendung der aufgeführten Quellen und Hilfsmittel angefertigt habe. Berlin, den 17. Oktober 2014 Rafael Reisenhofer Contents List of Figures 5 1 Introduction 7 2 Complex Shearlet Transforms 9 2.1 From Fourier to Shearlets via an Analysis of Optimality . ........ 10 2.1.1 FourierAnalysis........................... 14 2.1.2 Wavelets............................... 20 2.1.2.1 Wavelets in Two Dimensions . 26 2.1.3 Shearlets............................... 27 2.2 Wavelet and Shearlet Transforms . 33 2.3 Complex Wavelet and Shearlet Transforms . ... 34 3 Complex Shearlet Transforms and the Visual Cortex 41 4 Applications 45 4.1 EdgeDetection ............................... 45 4.1.1 PhaseCongruency ......................... 47 4.1.2 Detection of Singularities With Pairs of Symmetric and Odd- SymmetricWavelets ........................ 51 4.1.3 Edge Detection With Pairs of Symmetric and Odd-Symmetric Shearlets............................... 56 4.2 ImageQualityAssessment . 62 4.2.1 Structural Similarity Index (SSIM) . 63 4.2.2 Complex Shearlet-Based Image Similarity Measure . .... 67 4.2.3 NumericalResults ......................... 71 5 Discussion 75 A Some Definitions and Formulas 79 A.1.1 PolynomialDepthSearch . 81 References 83 3 4 List of Figures 2.1 A piecewise smooth function and a cartoon like image . ...... 11 2.2 ExampleofFouriermodes.
    [Show full text]
  • Operators and Frames
    OPERATORS AND FRAMES JAMESON CAHILL, PETER G. CASAZZA AND GITTA KUTYNIOK Abstract. Hilbert space frame theory has applications to various areas of pure mathematics, applied mathematics, and engineering. However, the ques- tion of how applying an invertible operator to a frame changes its properties has not yet been satisfactorily answered, and only partial results are known to date. In this paper, we will provide a comprehensive study of those questions, and, in particular, prove characterization results for (1) operators which gen- erate frames with a prescribed frame operator; (2) operators which change the norms of the frame vectors by a constant multiple; (3) operators which generate equal norm nearly Parseval frames. 1. Introduction To date, Hilbert space frame theory has broad applications in pure mathe- matics, see, for instance, [11, 8, 5], as well as in applied mathematics, computer science, and engineering. This includes time-frequency analysis [15], wireless communication [16, 20], image processing [19], coding theory [21], quantum mea- surements [14], sampling theory [13], and bioimaging [18], to name a few. A fundamental tool in frame theory are the analysis, synthesis and frame operators associated with a given frame. A deep understanding of these operators is indeed fundamental to frame theory and its applications. Let us start by recalling the basic definitions and notions of frame theory. M Given a family of vectors ffigi=1 in an N-dimensional Hilbert space HN , we say M that ffigi=1 is a frame if there exist constants 0 < A ≤ B < 1 satisfying that M 2 X 2 2 Akfk ≤ jhf; fiij ≤ Bkfk for all f 2 HN : i=1 The numbers A; B are called lower and upper frame bounds (respectively) for M the frame.
    [Show full text]
  • The Use of Adaptive Frame for Speech Recognition
    EURASIP Journal on Applied Signal Processing 2001:2, 82–88 © 2001 Hindawi Publishing Corporation The Use of Adaptive Frame for Speech Recognition Sam Kwong Department of Computer Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Email: [email protected] Qianhua He Department of Electronic Engineering, South China University of Technology, China Email: [email protected] Received 19 January 2001 and in revised form 18 May 2001 We propose an adaptive frame speech analysis scheme through dividing speech signal into stationary and dynamic region. Long frame analysis is used for stationary speech, and short frame analysis for dynamic speech. For computation convenience, the feature vector of short frame is designed to be identical to that of long frame. Two expressions are derived to represent the feature vector of short frames. Word recognition experiments on the TIMIT and NON-TIMIT with discrete Hidden Markov Model (HMM) and continuous density HMM showed that steady performance improvement could be achieved for open set testing. On the TIMIT database, adaptive frame length approach (AFL) reduces the error reduction rates from 4.47% to 11.21% and 4.54% to 9.58% for DHMM and CHMM, respectively. In the NON-TIMIT database, AFL also can reduce the error reduction rates from 1.91% to 11.55% and 2.63% to 9.5% for discrete hidden Markov model (DHMM) and continuous HMM (CHMM), respectively. These results proved the effectiveness of our proposed adaptive frame length feature extraction scheme especially for the open testing. In fact, this is a practical measurement for evaluating the performance of a speech recognition system.
    [Show full text]
  • Density, Overcompleteness, and Localization of Frames
    The Journal of Fourier Analysis and Applications Volume 12, Issue 2, 2006 Density, Overcompleteness, and Localization of Frames. I. Theory Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau Communicated by Hans G. Feichtinger ABSTRACT. 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 F = ffigi2I and E = fej gj2G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a: I ! G. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of F, the relative measure of E | 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 a(I) in G. 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 ap- proximated by elements of another frame.
    [Show full text]
  • FRAMES for BANACH SPACES Peter G. Casazza, Deguang Han
    FRAMES FOR BANACH SPACES Peter G. Casazza, Deguang Han and David R. Larson Abstract. We use several fundamental results which characterize frames for a Hilbert space to give natural generalizations of Hilbert space frames to general Ba- nach spaces. However, we will see that all of these natural generalizations (as well as the currently used generalizations) are equivalent to properties already extensively developed in Banach space theory. We show that the dilation characterization of framing pairs for a Hilbert spaces generalizes (with much more effort) to the Banach space setting. Finally, we examine the relationship between frames for Banach spaces and various forms of the Banach space approximation properties. We also consider Hilbert space frames. Here we classify the alternate dual frames for a Hilbert space frame by a natural manifold of operators on the Hilbert space. Most of the basic properties of alternate dual frames follow immediately from this characterization. We also answer a question of Han and Larson by showing that the dual frame pairs are compressions of Riesz basis and their dual bases. We also show that a family of frames for the same Hilbert space have the simultneous dilation property if and only if they have the same deficiency. 1. Introduction and background results In 1946 D. Gabor [18] introduced a fundamental approach to signal decomposition in terms of elementary signals. In 1952, while addressing some difficult problems from the theory of nonharmonic Fourier series, Duffin and Schaeffer [12] abstracted Gabor’s method to define frames for a Hilbert space. For some reason the work of Duffin and Schaeffer was not continued until 1986 when the fundamental work of Daubechies, Grossman and Meyer [11] brought this all back to life, right at the dawn of the “wavelet era”.
    [Show full text]