Introduction to Finite Frame Theory - Frame Research Center

More documents

Recommendations

Info

$Fast Algorithms for Signal Reconstruction without ... - Math Department$

8 Peter G. Casazza, Gitta Kutyniok, and Friedrich Philipp (b) The operator T is called injective (or one-to-one), if kerT = {0}, and surjective (or onto), if ranT = H K . It is called bijective (or invertible), if T is both injective and surjective. (c) The adjoint operator T ∗ : H K → H N is defined by (d) The norm of T is defined by 〈T x, ˜x〉 = 〈x,T ∗ ˜x〉 for all x ∈ H N and ˜x ∈ H K . ‖T ‖ := sup{‖T x‖ : ‖x‖ = 1}. The next result states several relations between these notions. Proposition 3. (i) Let T : H N → H K be a linear operator. Then dimH N = N = dimkerT + rankT. Moreover, if T is injective, then T ∗ T is also injective. (ii) Let T : H N → H N be a linear operator. Then T is injective if and only if it is surjective. Moreover , kerT = (ranT ∗ ) ⊥ , and hence H N = kerT ⊕ ranT ∗ . If T : H N → H N is an injective operator, then T is obviously invertible. If an operator T : H N → H K is not injective, we can make T injective by restricting it to (kerT ) ⊥ . However, T | (kerT ) ⊥ might still not be invertible, since it does not need to be surjective. This can be ensured by considering the operator T : (kerT ) ⊥ → ranT , which is now invertible. The Moore-Penrose inverse of an injective operator provides a one-sided inverse for the operator. Definition 6. Let T : H N → H K be an injective, linear operator. The Moore- Penrose inverse of T , T † , is defined by T † = (T ∗ T ) −1 T ∗ . It is immediate to prove invertibility from the left as stated in the following result. Proposition 4. If T : H N → H K is an injective, linear operator, then T † T = Id. Thus, T † plays the role of the inverse on ranT – not on all of H K . It projects a vector from H K onto ranT and then inverts the operator on this subspace. A more general notion of this inverse is called the pseudo-inverse, which can be applied to a non-injective operator. It, in fact, adds one more step to the action of T † by first restricting to (kerT ) ⊥ to enforce injectivity of the operator followed by application of the Moore-Penrose inverse of this new operator. This pseudoinverse can be derived from the singular value decomposition. Recalling that by fixing orthonormal bases of the domain and range of a linear operator we derive an
Introduction to Finite Frame Theory 9 associated unique matrix representation, we begin by stating this decomposition in terms of a matrix. Theorem 1. Let A be an M × N matrix. Then there exist an M × M unitary matrix U (see Definition 9), an N × N unitary matrix V , and an M × N diagonal matrix Σ with nonnegative, decreasing real entries on the diagonal such that A = UΣV ∗ . Hereby, an M × N diagonal matrix with M ≠ N is an M × N matrix (a i j ) M , N i=1, j=1 with a i j = 0 for i ≠ j. Definition 7. Let A be an M ×N matrix, and let U,Σ, and V be chosen as in Theorem 1. Then A = UΣV ∗ is called the singular value decomposition (SVD) of A. The column vectors of U are called the left singular vectors, and the column vectors of V are referred to as the right singular vectors of A. The pseudo-inverse A + of A can be deduced from the SVD in the following way. Theorem 2. Let A be an M × N matrix, and let A = UΣV ∗ be its singular value decomposition. Then A + = V Σ + U ∗ , where Σ + is the N × M diagonal matrix arising from Σ ∗ by inverting the non-zero (diagonal) entries. 2.2.2 Riesz Bases In the previous subsection, we already recalled the notion of an orthonormal basis. However, sometimes the requirement of orthonormality is too strong, but uniqueness of a decomposition as well as stability shall be retained. The notion of a Riesz basis, which we next introduce, satisfies these desiderata. Definition 8. A family of vectors (ϕ i ) N i=1 in a Hilbert space H N is a Riesz basis with lower (respectively, upper) Riesz bounds A (resp. B), if, for all scalars (a i ) N i=1 , we have ∥ N A∑ |a i | 2 N ∥∥∥∥ 2 N ≤ ∥∑ a i ϕ i ≤ B∑ |a i | 2 . i=1 i=1 i=1 The following result is immediate from the definition. Proposition 5. Let (ϕ i ) N i=1 be a family of vectors. Then the following conditions are equivalent. (i) (ϕ i ) N i=1 is a Riesz basis for H N with Riesz bounds A and B. (ii) For any orthonormal basis (e i ) N i=1 for H N , the operator T on H N given by Te i = ϕ i for all i = 1,2,...,N is an invertible operator with ‖T ‖ 2 ≤ B and ‖T −1 ‖ −2 ≥ A.
Page 1 and 2: Introduction to Finite Frame Theory
Page 7: Introduction to Finite Frame Theory
Page 51: Introduction to Finite Frame Theory

Introduction to Finite Frame Theory - Frame Research Center

Create successful ePaper yourself

Delete template?

Save as template?