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

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1 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 sufcient conditions on real sequences a a2 a M 0 so that there is a frame f' n g for H N with frame operator S and k' n k = a n, for all n =; 2;:::M. As a consequence we see that for any frame operator S on H N and for any M N, there is an equal norm frame for H N with M elements and having S as its frame operator. AMATLAB toolbox [4] implementing all results is freely distributed by the authors.. Introduction A sequence f' n g is a frame for an N-dimensional Hilbert space H N if the positive selfadjoint frame operator S = h'; ' n i' n is a bounded, invertible operator on H N. A frame f' n g is a -tight frame if S = I and if =, it is a Parseval frame. Moreover, Tr S = P k' n k 2. Hilbert space frames have played a fundamental role in signal/image processing since the seminal work of Gabor [7]. The tools introduced by Gabor where formalized into the notion of frames by Dufn and Schaeffer [5]. Recently, frames have been applied in a wide variety of areas from the Internet [8] and [9], multiple antenna coding [0], quantum theory [6], and [] and more. Each application of frame theory requires a new class of frames designed for the specic application. This often involves having to nd frames with ( prescribed in advance ) norms for the frame vectors. In [2] there is given necessary and sufcient conditions on real sequences a a 2 a M > 0 so that there exists a tight frame f' n g for H N with k' n k = a n, for all n =; 2;:::M. The condition for the existence of a -tight frame given in[2]isthat = a 2 n Na2 : One interpretation of this result is that it gives necessary and sufcient conditions on k' n k for f' n g to form a frame for H N with frame operator S = I. An alternative proof of this result appears in [3] where an algorithm is given for this construction which runsvery efciently in MATLAB. Date: April 5, Mathematics Subject Classication. 42C5. Key words and phrases. Finite tight frame, orthogonal matrix. P.G. Casazza and M. T. Leon were supported by NSF DMS

2 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 operator S on H N. That is, for a given S and M N, we give necessary and sufcient conditions on a a 2 a M > 0 so that there is a frame f' n g for H N with frame operator S and satisfying: k' n k = a n, for all n =; 2;:::M. 2. Main Result The main result in this paper is: Theorem 2.. Let S be a positive self-adjoint operator on a N-dimensional Hilbert space H N. Let 2 ::: N be the eigenvalues of S. Fix M N and real numbers a a 2 :::a M > 0. The following are equivalent: () There is a frame f' j g j=m ; 2;:::;M. for H N with frame operator S and k' j k = a j, for all j = (2) For every» k» N, (2.) i=k i=k a 2 i» i ; and It is well known ( see e.g. [] ) that there are equal norm Parseval frames with M-elements in H N for all M N. As a consequence of Theorem ( 2. ), we see that there are equal norm tight frames for any prescribed frame operator. Corollary 2.2. If S is a positive self-adjoint operator on H N then for every M N there is an equal norn frame f' n g for H N whose frame operator equals S. Proof. Let 2 N be the eigenvalues of S and let = i=n i. Fix M N. If P i=m a 2 i = i=n p a n = =M for every n then i=m a 2 n =, and for all» k» N, i=k a 2 n = k M = k M i=n i = k N M i=n N i» k N M i : i=k k In the next to last inequality above we have used the fact that deleting some of the smallest numbers from a set of numbers will increase the average of the numbers. p Hence, by Theorem ( 2. ), there is a frame f' n g with k' n k = =M for all n = ; 2;:::; M having frame operator S. To show that () implies (2) in the theorem we will actually prove a more general result. Theorem 2.3. Let f' j g j=m be a frame for H N with frame operator S having eigenvalues 2 N. If P is an orthogonal projection of H N onto a k-dimensional subspace,» k» N, then j=n j=n k+ j» j=m Proof. Let fe n g n=n be an orthonormal basis for H N with Se n = n e n, n = ; 2;:::;N. Let P be a rank k orthogonal projection on H N and let fψ i g i=k be an orthonormal basis for PH N. It is known ( see e.g. [2] ),that i» P i=k i :

3 () P P (2) FRAMES WITH A GIVEN FRAME OPERATOR 3 jh' n ;e m ij 2 = m, for all» m» N h' n ;e l ih' n ;e m i =0for all» l 6= m» N. Now we compute using () and (2) above. m= i=k i=k m= hψ i ;e m ih' n ;e m i kp' n k 2 = 2 = jhψ i ;e m ij 2 jh' n ;e m ij 2 + i=k i=k l=n m= l= i=k m6=l i=k jhψ i ;e m ij 2 m= jhψ i ;P' n ij 2 = i=k jhψ i ;' n ij 2 = hψ i ;e m ih' n ;e m ihψ i ;e l ih' n ;e l i = hψ i ;e m ihψ i ;e l i jh' n ;e m ij 2 = Since fψ i g i=k is an orthonormal basis for its span, we have that and i=k i=k m= n=n i=k m= jhψ i ;e m ij 2» ; for all» i» k;» m» N jhψ i ;e m ij 2 = i=k m= Combined with our calculations above, we obtain, ψ m=k i=k m m= m= jhψ i ;e m ij 2! m = jhψ i ;e m ij 2 = m= i=k kp' m k 2 h' n ;e l ih' n ;e m i = kψk 2 = k: k jhψ i ;e m ij 2 m : We now give two corollaries. The rst is one of the implications of Theorem (2.). Corollary 2.4. Let f' j g M be a frame for H N with frame operator S having eigenvalues 2 ::: N > 0. If k' k k' 2 k k' M k, then for every» k» N, k' j k 2» j Proof. Given k, let P be an orthogonal projection of rank k on H N so that ' j 2 PH N, for all» j» k. By Theorem ( 2.3 ) we have : k' j k 2 = j=m m :

4 4 PETER G. CASAZZA, MANUEL T. LEON The next corollary is well-known in many areas of mathematics. For example, in PDE's this is a consequence of the Rayleigh min/max principle. In stochastic processes this is the Karhunen- Loéwe theorem. Corollary 2.5. Let S be a positive self-adjoint operator on H N with eigenvalues 2 ::: N > 0. If P is a rank k orthogonal projection on H N then Tr(PSP)» Proof. If fe j g N is an orthonormal sequence in H N with Se j = j e j,thenf' j = p j e j g N is a frame for H N with frame operator S. Hence, PSP is the frame operator for fp' j g N. Applying Theorem ( 2.3 ), for every» k» N, we have Tr(PSP)= j=n Now we complete the proof of the main result by showing that (2) implies (). We'll start with aframef' j g j=m with frame operator S. The vectors used in Corollary ( 2.5 ) can be extended to a frame on H N with frame operator S. More generally, sinces is symmetric, S = V V where V is an orthonormal matrix and is a diagonal matrix with diag() = ( ; 2 ;:::; N ). Let M N be such that its top N rows equal 2 and all remaining entries are zero. Let W M M be orthonormal ( or unitary ), and let ' j = j th row of F = W V. Then The Gram operator is given by G = FF, and F F =(W V ) W V = V V = S: diag(g) =(k' k 2 ;:::;k' M k 2 ): Then ( Horn. p. 0) there is an orthogonal matrix U M M and an diagonal matrix such that G = U U; where diag() = ( ;:::; N ; 0;:::;0): Let V M M be (see Proposition ( 3. ) in the appendix ) an orthogonal matrix such that if T = V V ; then; diag(t )=(a 2 ;:::;a2 M): Let ψ j = j th row ofh = VUF. Then fψ j g 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 =(VUF) VUF = F F = S; diag(vuf(vuf) )=diag(vuff U V )=diag(v V )=(a 2 ;:::;a2 M):

5 FRAMES WITH A GIVEN FRAME OPERATOR 5 3. Appendix Every matrix in O(M) ( the orthogonal group ) is obtained as a product of Givens rotations (t; j; k; M) 2 O(M);j <k, where It is clear that (t; j; k; M) = 0 I j ;j cos(t) 0 sin(t) I M j k 2;M j k sin(t) 0 cos(t) I k ;k (t; j; k; M) = ( t; j; k; M) Proposition 3.. Let ;:::; M and a ;:::;a M be real numbers such that a 2 a2 2 a2 M and for every» k» M, (3.) i=k i=k a 2 i» i ; and Let be a diagonal matrix with diag() = ( ;:::; M ). Then there is a matrix O 2 O(M) such that i=m a 2 i = diag(oo )=(a 2 ;:::;a2 M): p Proof. We'll prove the proposition by induction on M. If M =2,lett = arcsin( a 2= 2 and O = (t; ; 2; 2). Next, Assume the result holds for M. From the hypothetsis, a 2. Let k be such that j a 2 for j =;:::;k and a 2 k. Let Then t = arcsin( q i=m i : a 2 = k and O = (t; ;k;m): O O = 0 a 2 0 ::: 0 :::0 0 :::.. ::: 0 :::.. 0 ::: where represents a possibly nonzero entry on kth row and st column ( or st row and kth column ). Let be the (M ) (M ) bottom right boxofo O. Then, is a diagonal matrix and, since Tr() = Tr(O O ), C A diag( )=( 2 ;:::; k ; k + a 2 ; k+;:::; M ): Now we'll verify that diag( ) and a 2 ;:::;a M meet the premises of the lemma. If m<k, m (m ) a 2 (m ) a2 2 a a 2 m : : C A

6 6 PETER G. CASAZZA, MANUEL T. LEON If m k, Then m = k + k + a 2 + k+ + + m = O = m a 2 a a 2 m : 0 0 O 2 O : where O 2 is the solution for, will satisfy the claim. References [] P. G. Casazza and J. Kova»cević. Equal norm tight frames with erasures. Advances in Computational Mathematics, special issue on frames, 2002.Invited Paper. [2] P. G. Casazza, J. Kova»cević, M. T. Leon, and J. C. Tremain. Custom built tight frames Preprint. [3] P. G. Casazza and M. T. Leon. Existence and construction of nite tight frames Preprint. [4] P. G. Casazza and M. T. Leon. Frameware [5] R. J. Dufn and A. C. Schaeffer. A class of nonharmonic fourier series. Trans. AMS, 72:34 366, 952. [6] Y. Eldar and G.D. Forney. Optimal tight frames and quantum measurement Preprint. [7] D. Gabor. Theory of communications. Jour. Inst. Elec. Eng. (London), 93: , 946. [8] V. K. Goyal and J. Kova»cević. Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal., 0(3): , 200. To appear. [9] V. K. Goyal, J. Kova»cević, and M. Vetterli. Multiple description transform coding: Robustness to erasures using tight frame expansions. Proc. IEEE Int. Symp. on Inform. Th., 998. Cambridge, MA. [0] B. Hochwald, T. Marzetta, T. Richardson, W. Sweldens, and R. Urbanke. Systematic design of unitary space-time constellations. IEEE Trans. Inform. Th., Submitted. [] E.»Soljanin. Tight frames in quantum information theory. DIMACS Workshop on Source Coding and Harmonic Analysis. Department of Mathematics, University of Missouri, Columbia, MO 652 address: pete@math.missouri.edu,mleon@math.missouri.edu

University of Missouri Columbia, MO USA

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 65211 USA e-mail: casazzap@missouri.edu