University of Missouri Columbia, MO USA

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1 EXISTENCE AND CONSTRUCTION OF FINITE FRAMES WITH A GIVEN FRAME OPERATOR PETER G. CASAZZA 1 AND MANUEL T. LEON 2 1 Department of Mathematics University of Missouri Columbia, MO USA casazzap@missouri.edu 2 Department of Mathematics University of Missouri Columbia, MO USA casazzap@missouri.edu Abstract. Let S be a positive self-adjoint invertible operator on an N-dimensional Hilbert space H N and let M N. We give necessary and sufficient conditions on real sequences a 1 a 2 a M 0 so that there is a frame {ϕ n } M n1 for H N with frame operator S and ϕ n a n, for all n 1, 2,... M. As a consequence, given any frame operator S as above, there is a set of equal norm vectors in H N which have precisely S as their frame operator. 1. Introduction A sequence {ϕ n } M n1 is a frame for an N-dimesional Hilbert space H N if the positive self-adjoint frame operator S ϕ, ϕ m ϕ m is an invertible operator on H N. A frame {ϕ n } nm n1 is a λ-tight frame if S λi and if λ 1, it is a Parseval frame. If the frame vectors all have the same norm, this is an equal-norm frame. The analysis operator of the frame is T : H N l 2 (M) given by: T (ϕ) { ϕ, ϕ m } M. The synthesis operator is T where T ({a m } M ) M a mϕ m. So the frame operator is S T T. The Gram matrix is the matrix of the operator T T. For an introduction to frame theory, see [1, 5, 6]. In [2] there is given necessary and sufficient conditions on real sequences a 1 a 2 a M > 0 so that there exists a tight frame {ϕ m } M for H N with ϕ m a m, for all n 1, 2,... M. The condition for the existence of a λ-tight frame given in [2] is that λ a 2 m a Mathematics Subject Classification. 47A05,47A10. Key words and phrases. Finite tight frame, orthogonal matrix. 1

2 One interpretation of this result is that it gives necessary and sufficient conditions on ϕ m for {ϕ m } M to form a frame for H N with frame operator S λi. An alternative constructive proof of this result appears in [4] where an algorithm is given for this construction. In this paper we generalize these results to the case where λi is replaced by any positive selfadjoint invertible operator S on H N. That is, for a given S and M N, we give necessary and sufficient conditions on a 1 a 2 a M > 0 so that there is a frame {ϕ m } M for H N with frame operator S and satisfying: ϕ m a m, for all m 1, 2,... M. We will then see that every frame operator S can be realized as the frame operator of an equal norm frame with M-elements, for any M N. The main result in this paper is: 2. Main Result Theorem 2.1. Let S be a positive self-adjoint operator on a N-dimensional Hilbert space H N. Let λ 1 λ 2... λ N > 0 be the eigenvalues of S. Fix M N and real numbers a 1 a 2 a M > 0. The following are equivalent: (1) There is a frame {ϕ j } M for H N with frame operator S and ϕ j a j, for all j 1, 2,..., M. (2) For every 1 k N, (2.1) a 2 i λ i, and a 2 i λ i. Now we proceed to prove Theorem 2.1. To show that (1) implies (2) in the theorem we will actually prove a more general result. Theorem 2.2. Let {ϕ j } jm be a frame for H N with frame operator S having eigenvalues λ 1 λ 2 λ N. If P is an orthogonal projection of H N onto a k-dimensional subspace of H N, 1 k N, then jn k+1 λ j P ϕ j 2 λ j. Proof. Let {e n } nn n1 be an orthonormal basis for H N with Se n λ n e n, n 1, 2,..., N. Let P be a rank k orthogonal projection on H N and let {ψ i } k be an orthonormal basis for P H N. It is known ( see e.g. [3, 5] ),that (1) M ϕ n, e m 2 λ m, for all 1 m N n1 (2) M ϕ n, e l ϕ n, e m 0 for all 1 l m N. n1

3 Now we compute P ϕ n 2 n1 n1 n1 n1 ψ i, P ϕ n 2 ψ i, ϕ n 2 ψ i, e m ϕ n, e m n1 l1 n1 ψ i, e m ϕ n, e m ψ i, e l ϕ n, e l ψ i, e m 2 ϕ n, e m 2 + ψ i, e m 2 M n1 ψ i, e m 2 λ m. 2 ϕ n, e m 2 ψ i, e m ψ i, e l ϕ n, e l ϕ n, e m m l Since {ψ i } k is an orthonormal basis for its span, we have that and ψ i, e m 2 1, for all 1 i k, 1 m N ψ i, e m 2 ψ i, e m 2 Hence (See Lemma 4.1 in the Appendix), ( ) λ m ψ i, e m 2 λ m ψ i 2 k. P ϕ m 2 mn k We now give two corollaries. The first is the implication (1) (2) of Theorem 2.1. Corollary 2.3. Let {ϕ j } M be a frame for H N with frame operator S having eigenvalues λ 1 λ 2... λ N > 0. If ϕ 1 ϕ 2 ϕ M, then for every 1 k N, n1 λ m. ϕ j 2 λ j

4 Proof. Given k, let P be an orthogonal projection of rank k on H N so that ϕ j P H N, for all 1 j k. By Theorem 2.2 we have : ϕ j 2 P ϕ j 2 P ϕ j 2 λ j. Corollary 2.4. Let S be a positive self-adjoint operator on H N λ N > 0. If P is a rank k orthogonal projection on H N then T r(p SP ) λ j. with eigenvalues λ 1 λ 2 Proof. If {e j } N is an orthonormal sequence in H N with Se j λ j e j, then {ϕ j λ j e j } N is a frame for H N with frame operator S. Hence, P SP is the frame operator for {P ϕ j } N. Applying Theorem 2.2, for every 1 k N, we have T r(p SP ) P ϕ j 2 λ j. 3. Equal Norm Frames We ll start with a frame {ϕ j } M with frame operator S. The vectors used in Corollary 2.4 can be extended to a frame on H N with frame operator S. More generally, since S is symmetric, S V ΛV where V is a unitary matrix and Λ is a diagonal matrix with diag(λ) λ 1, λ 2,..., λ N. Let M N be such that its top N rows equal Λ 1 2 and all remaining entries are zero. Let W M M be unitary, and let ϕ j j th row of F W V. Then The Gram operator is given by G F F, and F F (W V ) W V V ΛV S diag(g) ( ϕ 1 2,..., ϕ M 2 ). Then (See e.g. [7]) there is an orthogonal matrix U M M and an diagonal matrix Λ such that G U ΛU, where diag(g) (λ 1,..., λ N, 0,..., 0). Let V M M be (see Lemma 4.2 in the appendix ) an orthogonal matrix such that if T V ΛV, then, diag(t ) (a 2 1,..., a 2 M). Let ψ j j th row of H V UF. Then {ψ j } M is a frame, since rank(h) rank(f ) N. Its frame operator is given by and the diagonal of its Gram matrix is H H (V UF ) V UF F F S, diag(v UF (V UF ) ) diag(v UF F U V ) diag(v ΛV ) (a 2 1,..., a 2 M).

5 Now we check that the requirements of Theorem 2.1 are always met for equal norm frames. Corollary 3.1. Let S be a positive self-adjoint operator on a N-dimensional Hilbert space H N. For any M N there is an equal norm sequence {ϕ m } M in H N which has S as its frame operator. Proof. Let λ 1 λ 2... λ N 0 be the eigenvalues of S. Let (3.1) a 2 1 λ i. M Now we check the conditions of Theorem 2.1 (2) to see that there is a sequence {ϕ m } M in H N with f m a for all m 1, 2,..., M and having frame operator S. We are letting a 1 a 2 a M a. For the second equality in Theorem 2.1, by Equation 3.1, (3.2) ϕ m 2 a 2 m Ma 2 λ i. For the first inequality in Theorem 2.1, we note that by Equation 3.1 we have that a 2 1 a 2 1 λ i 1 λ i λ 1. M N So our inequality holds for m 1. Suppose there is an 1 < m N for which this inequality fails, and m is the first time this fails, and we will come to a contradiction. So, while It follows that Hence, m 1 Ma 2 But this contradicts Equation 3.2. a 2 i (m 1)a 2 m a 2 i ma 2 > m 1 m λ i. λ i, a 2 m a 2 > λ m λ m+1 λ N. a 2 m > m a 2 i + m λ i + m λ i + λ i. im+1 im+1 im+1 a 2 i a 2 i λ i

6 4. Proof of the Main Theorem Now we complete the proof of the main result by showing that (2) implies (1). Lemma 4.1. Assume we have two sets of numbers {c m } N and {λ m } N satisfying: 1. λ 1 λ 2 λ N. 2. We have 0 c m 1 and N c m k. Then λ m c m λ m mn k Proof. We will check the first inequality. The second will follow similarly. Choose non-negagtive numbers {b m } k so that c m + b m 1, for all m 1, 2,..., k. We first observe that That is, c m + mk+1 Since {λ m } N, we have c m c m k b m λ m. (c m + b m ) mk+1 c m. c m + b m. c m λ m ( N ) c m λ m + c m λ k+1 mk+1 ( ) c m λ m + b m λ k+1 (c m λ m + b m λ k+1 ) (c m λ m + b m λ m ) (c m + b m )λ m λ m.

7 Every matrix in O(M) ( the orthogonal group ) is obtained as a product of Givens rotations θ(t, j, k) O(M), j < k, where I j 1,j cos(t) 0 sin(t) 0 θ(t, j, k) 0 0 I M j k 2,M j k sin(t) 0 cos(t) I k 1,k 1 It is clear that θ(t, j, k) 1 θ( t, j, k) Lemma 4.2. Let λ 1,..., λ M and a 1,..., a M be real numbers such that a 2 1 a 2 2 a 2 M and for every 1 k M, (4.1) a 2 i λ i, and a 2 i λ i. Let Λ be a diagonal matrix with diag(λ) (λ 1,..., λ M ). Then there is a matrix O O(M) such that diag(oλo ) (a 2 1,..., a 2 M). Proof. We ll prove the lemma by induction on M. If M 2, let t arcsin( λ 2 a 2 2/λ 2 a 2 1 and O θ(t, 1, 2, 2), and we are done. Now, assume the result holds for M 1. From the hypothetsis, λ 1 a 2 1, let k be such that λ j a 2 1 for j 1,..., k 1 and a 2 1 λ k. Let t arcsin( λ 1 a 2 1/λ 1 λ k and O 1 θ(t, 1, k, M). Then a O 1 ΛO Let Λ 1 be the (M 1) (M 1) bottom right box of O 1 ΛO 1. Then, Λ 1 is a diagonal matrix and, since T r(λ) T r(o 1 ΛO 1), diag(λ 1 ) (λ 2,..., λ k 1, λ k + λ 1 a 2 1, λ k+1,..., λ M ). Now we ll verify that Λ 1 and a 2,..., a M satisfy the hypotheses of the lemma. If m < k, If m k, λ 2 + λ λ m (m 1) a 2 1 (m 1) a 2 2 a a 2 m. λ 2 + λ λ m λ 2 + λ λ k 1 + λ k + λ 1 a λ k λ m λ 1 + λ λ m a 2 1 a a 2 m,

8 thus giving λ 1 + λ λ m a a a 2 m, Then, by our induction hypothesis, there is an O 2 O(M 1) such that Then will satisfy the claim diag(o 2 Λ 2 O 2) (a 2 2,..., a 2 M). O ( ) 1 0 O 0 O Acknowledgement P.G. Casazza was supported by NSF DMS , References [1] P.G. Casazza, The art of frame theory. Taiwanese J. Math., 4 (2000), [2] P.G. Casazza, M. Fickus, J. Kovačević, and J.C. Tremain, A physical interpretation for finite tight frames, in Harmonic Analysis and Applications (In Honor of John Benedetto), C. Heil, Ed. Birkhauser (2006), [3] P.G. Casazza and J. Kovačević, Uniform tight frames with erasures, Advances in Computational Math. 18, No s 2-4 (2003), [4] P.G. Casazza and M. Leon, Existence and construction of finite tight frames, Journal of Cocrete and applicable mathematics, 4 No. 3 (2006), [5] P.G. Casazza and J.C. Tremain, A short introduction to frame theory, preprint ( [6] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, Boston (2003). [7] R.A. Horn and C.R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, U.K. (1999).

2 PETER G. CASAZZA, MANUEL T. LEON In this paper we generalize these results to the case where I is replaced by any positive selfadjoint invertible op

2 PETER G. CASAZZA, MANUEL T. LEON In this paper we generalize these results to the case where I is replaced by any positive selfadjoint invertible op FRAMES WITH A GIVEN FRAME OPERATOR PETER G. CASAZZA, MANUEL T. LEON Abstract. Let S be a positive self-adjoint invertible operator on an N-dimensional Hilbert space H N and let M N. We give necessary and