Frames in finite-dimensional spaces Ole Christensen Department of Applied Mathematics and Computer Science Harmonic Analysis - Theory and Applications (HATA DTU) Technical University of Denmark ochr@dtu.dk July 27, 2015 (DTU) Talk, Bremen, 2015 July 27, 2015 1 / 47
• An Introduction to frames and Riesz bases, Birkh¨ auser 2002. • Second expanded edition (720 pages), Spring 2016 • Chapter 1: Frames in finite-dimensional spaces. • If you want a pdf-file with Chapter 1 - contact me at ochr@dtu.dk (DTU) Talk, Bremen, 2015 July 27, 2015 2 / 47
Plan for the talk • Frames in finite-dimensional versus infinite-dimensional spaces; • (Explicit constructions of tight frames in C n with desirable properties) (Talks by Fickus, Mixon, Strawn) • Tight frames versus dual pairs of frames in C n ; • Gabor frames in L 2 ( R ) and dual pairs; • From Gabor frames in L 2 ( R ) to Gabor frames in C n through sampling and periodization. (Talk by Malikiosis) • 6 open problems along the way. (DTU) Talk, Bremen, 2015 July 27, 2015 3 / 47
Key purpose of frame theory Let V denote a vector space. Want: Expansions � f = c k f k of signals f ∈ V in terms of convenient building blocks f k . Desirable properties could be: • Easy to calculate the coefficients c k ; • Only few large coefficients c k for the relevant signals f ; • Stability against noise or removal of elements. The vector space can be • A finite-dimensional vector space with inner product, typically R n or C n ; • An infinite-dimensional Hilbert space; either an abstract space, or a concrete space, typically L 2 ( R ) , ℓ 2 ( Z ) , or L 2 ( 0 , L ) . • A Banach space or a topological space ( L p ( R ) , Besov spaces, modulation spaces, Fr´ echet spaces) (DTU) Talk, Bremen, 2015 July 27, 2015 4 / 47
Four classical tracks in frame theory • Finite frames; • Frame theory in separable Hilbert spaces; • Gabor frames in L 2 ( R ); • Wavelet frames in L 2 ( R ); • (Geometric analysis: curvelets, shearlets,......) • (Frames in Banach spaces, abstract generalizations, Hilbert C ∗ modules,.....). To a large extent the 4 topics are developed independently of each other - but more coordination would be useful! (DTU) Talk, Bremen, 2015 July 27, 2015 5 / 47
Frames - a generalization of orthonormal bases Definition: Let H denote a Hilbert space. A family of vectors { f k } k ∈ I is a frame for H if there exist constants A , B > 0 such that � A || f || 2 ≤ |� f , f k �| 2 ≤ B || f || 2 , ∀ f ∈ H . k ∈ I The numbers A , B are called frame bounds. The frame is tight if we can choose A = B . Note that (i) If H is an infinite-dimensional Hilbert space, the index I must be infinite; (ii) If H is finite-dimensional, the index set I can still be infinite (although in general not very natural) (DTU) Talk, Bremen, 2015 July 27, 2015 6 / 47
General frame theory Theorem Let { f k } k ∈ I be a frame for H . Then the following hold: (i) The operator � S : H → H , Sf := � f , f k � f k k ∈ I as well-defined, bounded, self-adjoint, and invertible; (ii) Each f ∈ H has the expansion � � f = 1 � f , S − 1 f k � f k f = � f , f k � f k Tight case: A k ∈ I k ∈ I (iii) If { f k } k ∈ I is a frame but not a basis, there exists families { g k } k ∈ I � = { S − 1 f k } k ∈ I such that � f = � f , g k � f k , ∀ f ∈ H . k ∈ I Any such { g k } ∞ k = 1 is called a dual frame. (DTU) Talk, Bremen, 2015 July 27, 2015 7 / 47
Frames in finite-dimensional spaces A frame for C n is a collection of vectors { f k } m k = 1 in C n such that there exists constants A , B > 0 with the property m � A || f || 2 ≤ |� f , f k �| 2 ≤ B || f || 2 , ∀ f ∈ C n . k = 1 k = 1 in C n is a frame if and only if Proposition A family of vectors { f k } m span { f k } m k = 1 = C n . k = 1 in C n is a frame for C n , then m ≥ n . Corollary If { f k } m Frame theory in C n is really “just” linear algebra! (DTU) Talk, Bremen, 2015 July 27, 2015 8 / 47
Frames in finite-dimensional spaces There are (at least) two tracks in frame theory in finite-dimensional spaces: (i) Explicit construction of frames with desired properties; (ii) Analysis of the interplay between frames in finite-dimensional spaces and in infinite-dimensional spaces. The focus in this talk will be on (ii). (DTU) Talk, Bremen, 2015 July 27, 2015 9 / 47
Bases and linear algebra Classical results from linear algebra in C n k = 1 in C n can be extended • Every set of linearly independent vectors { f k } m to a basis; i.e., there exist vectors { g k } ℓ k = 1 such that k = 1 ∪ { g k } ℓ { f k } m k = 1 is a basis for C n ; • Every family { f k } m k = 1 of vectors such that span { f k } m k = 1 = C n , contains a basis; that is, there exists an index set I such that { f k } k ∈{ 1 ,..., m }\ I is a basis for C n . (DTU) Talk, Bremen, 2015 July 27, 2015 10 / 47
Frames in finite-dimensional spaces Frame formulation: Proposition: k = 1 in C n can be extended to a (tight) (i) Every finite set of vectors { f k } m frame; i.e., there exist vectors { g k } ℓ k = 1 such that { f k } m k = 1 ∪ { g k } ℓ k = 1 is a (tight) frame for C n ; k = 1 for C n contains a basis; that is, there exists an index (ii) Every frame { f k } m set I such that { f k } k ∈{ 1 ,..., m }\ I is a basis for C n . (DTU) Talk, Bremen, 2015 July 27, 2015 11 / 47
Frame theory in infinite-dimensional spaces is different: Let H denote an infinite-dimensional separable Hilbert space. Theorem (Li/Sun, Casazza/Leonhard, 2008) Every finite set of vectors in H can be extended to a tight frame. (DTU) Talk, Bremen, 2015 July 27, 2015 12 / 47
Frame theory in infinite-dimensional spaces is different: Let H denote an infinite-dimensional separable Hilbert space. Theorem (Li/Sun, Casazza/Leonhard, 2008) Every finite set of vectors in H can be extended to a tight frame. Theorem (Casazza, C., 1995) There exist frames { f k } ∞ k = 1 , for which no subfamily { f k } k ∈ N \ I is a basis for H . Example Let { e k } ∞ k = 1 denote an ONB for H . Then the sequence � � e 1 , 1 e 2 , 1 e 2 , 1 e 3 , 1 e 3 , 1 { f k } ∞ k = 1 := √ √ √ √ √ e 3 , · · · 2 2 3 3 3 is a tight frame; but no subfamily is a Riesz basis. (DTU) Talk, Bremen, 2015 July 27, 2015 12 / 47
Frame theory in infinite-dimensional spaces is different: A much more complicated result: Proposition (Casazza, C., 1995) There exist tight frames { f k } ∞ k = 1 with || f k || = 1 , ∀ k ∈ N , for which no subfamily { f k } k ∈ N \ I is a basis for H . (DTU) Talk, Bremen, 2015 July 27, 2015 13 / 47
A sequence with a strange behavior Example (C., 2001) Let { e k } ∞ k = 1 denote an ONB for H and define { f k } ∞ k = 1 by f k := e k + e k + 1 , k ∈ N . Then (i) span { f k } ∞ k = 1 = H ; (ii) { f k } ∞ k = 1 is a Bessel sequence, but not a frame; (iii) There exists f ∈ H such that ∞ � f � = c k f k k = 1 for any choice of the coefficients c k . k = 1 is minimal and its unique biorthogonal sequence { g k } ∞ (iv) { f k } ∞ k = 1 is given by k � g k = ( − 1 ) k ( − 1 ) j e j , k ∈ N . j = 1 (DTU) Talk, Bremen, 2015 July 27, 2015 14 / 47
A classical ONB for C n Given n ∈ N , let ω := e 2 π i / n and consider the n × n discrete Fourier transform matrix (DFT) given by · · 1 1 1 1 ω 2 ω n − 1 1 ω · · ω 2 ω 4 ω 2 ( n − 1 ) · · 1 1 √ n . · · · · · 1 1 · · · · · ω n − 1 ω 2 ( n − 1 ) ω ( n − 1 )( n − 1 ) 1 · · (DTU) Talk, Bremen, 2015 July 27, 2015 15 / 47
A classical ONB for C n Given n ∈ N , consider the n vectors e k , k = 1 , . . . , n in C n , given by 1 e 2 π i k − 1 n e 4 π i k − 1 e k = 1 n √ n , k = 1 , . . . n . · · e 2 π i ( n − 1 ) k − 1 n Note that e k is the k th column in the Fourier transform matrix (DFT). Lemma: The vectors { e k } n k = 1 constitute an orthonormal basis for C n . (DTU) Talk, Bremen, 2015 July 27, 2015 16 / 47
Tight frames in C n - the first construction Construction by Zimmermann (2001), motivated by question by Feichtinger: k = 1 in C n by Theorem: Let m > n and define the vectors { f k } m 1 e 2 π i k − 1 m 1 √ m f k = · , k = 1 , 2 , . . . , m . · e 2 π i ( n − 1 ) k − 1 m k = 1 is a tight overcomplete frame for C n with frame bound equal to Then { f k } m one, and || f k || = � n m for all k . Note that the vectors f k consist of the first n coordinates of the Fourier ONB k = 1 in C n is called a harmonic frame. for C m . The frame { f k } m (DTU) Talk, Bremen, 2015 July 27, 2015 17 / 47
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