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Frame Theory Lecture 3 – Frame Theory David Walnut Department of Mathematical Sciences George Mason University Fairfax, VA USA Chapman Lectures, Chapman University, Orange, CA 6-10 November 2017 Walnut (GMU) Lecture 3 – Frame Theory Outline Finite Frames Abstract Frames The Frame operator Riesz bases Some historical remarks The BLT from the perspective of frames Applications of frames Walnut (GMU) Lecture 3 – Frame Theory Finite Frames and Linear Algebra We start from the elementary fact that a collection of d vectors in Cd , X = fx1; x2;:::; xd g is a basis for Cd if and only if X is linearly independent, that is, if c1x1 + c2x2 + ··· + cd xd = 0 =) c1 = c2 = ··· = cd = 0: Every x 2 Cd can be written uniquely as x = a1x1 + a2x2 + ··· + ad xd where ai = hx; xei i and Xe = fxe1; xe2;:::; xed g is the dual basis of X. Walnut (GMU) Lecture 3 – Frame Theory Matrix notation Let 2 j j j 3 B = 4 x1 x2 ··· xd 5 j j j Since X is linearly independent, B is invertible. 2 3 xe1 6 x 7 −1 6 e2 7 d B = 6 . 7 where fxj g is the dual basis. 6 . 7 e j=1 4 . 5 xed 2 3 hx; xe1i d 6 hx; xe2i 7 X x = BB−1x = B 6 7 = hx; x i x . 6 . 7 ej j 4 . 5 j=1 hx; xed i Walnut (GMU) Lecture 3 – Frame Theory Finite Frames d Let Y = fy1; y2;:::; yng ⊆ C , where n > d be a spanning set for Cd . We say then that Y is a frame for Cd . Since Y contains a basis for Cd , for all x 2 Cd there exist coefficients αj such that x = α1y1 + α2y2 + ··· + αnyn: However these coefficients need not be unique. Frames mimic bases but are redundant in the sense that Not all of the frame elements need be present in order to represent elements in the vector space, and There are multiple representations of each vector in the space. Walnut (GMU) Lecture 3 – Frame Theory Matrix notation Let F = y1 y2 ··· yn Since Y is a frame, F has full rank, the d × d matrix FF ∗ is invertible. For all x 2 Cd , x = FF ∗(FF ∗)−1x, and note that F y = F ∗(FF ∗)−1 is the pseudoinverse of F. Letting 2 3 2 3 ye1 hx; ye1i 6 7 hx; y i n y 6 ye2 7 y 6 e2 7 X F = 6 . 7 ; x = FF x = F 6 . 7 = hx; yej i yj : 6 . 7 6 . 7 4 . 5 4 5 j=1 hx; y i yen en Walnut (GMU) Lecture 3 – Frame Theory By properties of the pseudoinverse, c = F yx is the solution to Fc = x with the smallest norm. Hence of all coefficients α1; α2; : : : ; αn that satisfy n X x = αj yj ; j=1 αj = hx; yej i has minimal norm. n The collection of vectors fyej gj=1 is called the dual frame of Y . Walnut (GMU) Lecture 3 – Frame Theory Abstract frames Definition A frame in a separable Hilbert space H is a sequence of vectors fxk gk2K with the property that there exist constants A; B > 0, called the frame bounds such that for all x in the Hilbert space 2 X 2 2 A kxk ≤ jhx; xk ij ≤ B kxk : k2K A frame is tight if A = B and is uniform if xj = kxk k for all j and k. A tight frame with frame bounds A = B = 1 is a Parseval frame. Note that in the case of finite frames, this inequality is 2 ∗ 2 A kxkCd ≤ kF xkCn ≤ B kxkCd . This inequality is always satisfied if F has full rank. Walnut (GMU) Lecture 3 – Frame Theory The frame operator Definition Given a frame fxk g, we define the analysis operator by 2 T : H ! ` (K ); x 7! fhx; xk ig: Its adjoint is synthesis operator ∗ 2 X T : ` (K ) ! H; fck g 7! ck xk : The frame operator for fxk gk2K is ∗ X S = T T : H ! H; x 7! hx; xk i xk : Walnut (GMU) Lecture 3 – Frame Theory The analysis operator corresponds to the mapping F ∗ : Cd ! Cn, x 7! F ∗x, and the synthesis operator to F : Cn ! Cd , y 7! Fy. The frame inequality can be written as Ahx; xi ≤ hSx; xi ≤ Bhx; xi or as AI ≤ S ≤ BI so that S is a self-adjoint positive operator on H. Walnut (GMU) Lecture 3 – Frame Theory The frame operator After some manipulation, we arrive at the operator inequality A − B 2 B − A I ≤ I − S ≤ I: A + B B + A B + A This implies that S is an isomorphism of H and that 2 B − A I − S ≤ < 1: B + A B + A If the frame is tight (A = B) then S is a multiple of the identity. Numerical inversion of S converges very rapidly if A ≈ B, so that good frame bounds are important. Walnut (GMU) Lecture 3 – Frame Theory The dual frame We have the following representations of x 2 H. −1 X −1 X −1 x = SS x = hS x; xk i xk = hx; S xk i xk ; −1 X −1 x = S Sx = hx; xk i S xk : −1 The collection fS xk g is called the dual frame of fxk g. In general frames are redundant, that is, there exist 2 sequences c = fck g 2 ` n f0g such that ∗ X T c = ck xk = 0: Walnut (GMU) Lecture 3 – Frame Theory P For all sequences fck g such that x = ck xk , X 2 X −1 2 X −1 2 jcnj = jhx; S xnij + jhx; S xni − cnj : −1 This implies that of all such ck , ck = hx; S xk i has the smallest norm. Walnut (GMU) Lecture 3 – Frame Theory Riesz bases A frame that ceases to be a frame upon the removal of one element is called an exact frame. An exact frame satisfies the following. 2 The analysis operator T : H ! ` , x 7! fhx; xk ig is an isomorphism. The synthesis operator T ∗ : `2 ! H is injective, i.e., the frame is linearly independent. −1 The dual frame fS xk g is biorthogonal to fxk g. 2 For every sequence ck 2 ` , 2 −1 X 2 X −1 X 2 B jck j ≤ ck xk ≤ A jck j : We say in this case that fxk g is a Riesz basis for H. Walnut (GMU) Lecture 3 – Frame Theory Some historical remarks The notion of a frame was first introduced in 1952 by Duffin and Schaeffer in the context of nonharmonic Fourier series. The paper referenced results of Paley and Wiener (1930) on basicity properties of sets E(Λ) = fe2πiλx : λ 2 Λg in L2(−1=2; 1=2) where Λ ⊆ R is a perturbation of Z. Theorem (Duffin-Schaeffer) The collection E(Λ) is a frame for L2(−γ; γ) for all 0 < γ < 1=2 if the set Λ has uniform density 1, that is, if there exist constants δ; L > 0 such that for all n 2 Z, jλn − nj ≤ L and for all n 6= m, jλn − λmj ≥ δ (Λ is uniformly discrete). Walnut (GMU) Lecture 3 – Frame Theory Beurling density Uniform density is a special case of Beurling density (Landau, 1967). Given Λ ⊆ R, uniformly discrete, define for r > 0 n+(r) = sup jΛ \ [x; x + r]j and n−(r) = inf jΛ \ [x; x + r]j: 2 x2R x R n+(r) n−(r) Let D+(Λ) = lim and D−(Λ) = lim denote r!1 r r!1 r the upper and lower Beurling densities of Λ. If D+(Λ) = D−(Λ) then the common value D(Λ) is the Beurling density of Λ. Walnut (GMU) Lecture 3 – Frame Theory H. J. Landau (1967) showed, among other things, that, given a uniformly discrete subset Λ of R, E(Λ) is a frame for L2(−γ; γ), then D−(Λ) ≥ 2γ, and that if E(Λ) is a Riesz basis for L2(−γ; γ), then D+(Λ) ≤ 2γ. As a consequence, if Λ has uniform density 1, frames E(Λ) must necessarily be overcomplete for L2(−γ; γ) whenever 0 < γ < 1=2. Hence the Duffin and Schaeffer result describes necessarily redundant systems. Walnut (GMU) Lecture 3 – Frame Theory Most of the work on frames in the 60s and 70s focused on properties of non-redundant systems of exponentials. Daubechies, Grossman, and Meyer (1986) connected explicitly the notion of a frame with the expansion of a function in terms of so-called coherent states. By this was meant the image of a single function by a fixed collection of transformations based on the Weil-Heisenberg group (Gabor expansions) and the affine group (wavelets). Having a frame as opposed to a Riesz basis is necessary in order to have stable expansions in terms of atoms with desirable properties. Walnut (GMU) Lecture 3 – Frame Theory The BLT and frames Theorem 0 Given Ω; Ω > 0 and any function g(t) 2 L2(R), define 2πimΩ0t gn=Ω;mΩ0 (t) = g(t − n=Ω) e : 0 2 If Ω > Ω then fgn=Ω;mΩ0 gn;m2Z is incomplete in L (R) and hence not a frame. 0 If Ω < Ω and fgn=Ω;mΩ0 gn;m2Z is a frame, then it is not a Riesz basis, i.e., the frame is overcomplete. 0 If Ω = Ω and fgn=Ω;mΩ0 gn;m2Z is a frame, then it is a Riesz basis, i.e., it is exact and Z 1 Z 1 jt g(t)j2 dt · j! g^(!)j2 d! = 1: −∞ −∞ Walnut (GMU) Lecture 3 – Frame Theory Frames in Communication Theory Consider the following model for transmission of a signal over a channel.
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