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 .