Available online at http://ijim.srbiau.ac.ir Int. J. Industrial Mathematics Vol. 2, No. 1 (2010) 43-46
Comment on" Frames of Subspaces" H. R. Rahimi a , H. Youse Nejad b
(a) Department of Mathematics, Faculty of Science, Centeral Tehran Branch, Islamic Azad University, Tehran, Iran (b) Department of Mathematics, Faculty of Science, North Branch, Islamic Azad University, Tehran, Iran Received 25 September 2009; revised 4 February 2010; accepted 18 February 2010.
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In this paper we give a counter example for one of the lemmas of the paper" Frame of subspace in Wavelets, frames and operator theory" by P.G. Casazza and G. Kutyniok . Keywords : Hilbert space; Fusion Frame; Orthogonal Projection; Orthonormal basis.
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Introduction
Frames were rst introduced by Dun and Schae er[4] in the context of nonharmonic Fourier series, and today frames play important roles in many applications in mathematics, science, and engineering, including time-frequency analysis [5], internet coding [6], speech and music processing[11], communication [9], multiple antenna coding [8], medicine [10], quantum computing [7], and many other areas. For the discussion of the following section, we state here some de nitions, notations and known results. For convenience of readers, we suggest that one refer to [1, 2, 3] for details. Let H be a separable Hilbert space and let I be a countable (or nite) index set. If W is a closed subspace of H , we denote the orthogonal projection of H onto W by W . A sequence F = fPfi gi2I in H is a frame for H if there exist constants 0 < A B < 1 such that Akf k2 i2I j hf; fi ij2 B k f k2 for all f 2 H . The numbers A; B are called lower and upper frame bounds, respectively. The family F is called a tight frame if A = B , it is a Parseval frame if A = B = 1, it is a x-uniform frame if k fi k=k fj k= x for all i; j 2 I and an exact frame if it ceases to be a frame when any one of its elements is removed. If the right-handed of mentioned inequality holds, then we say that F is a Bessel sequence and call B the Bessel bound. The operator SF : H ! H is called frame Corresponding author. Email address:h
[email protected]
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H. R. Rahimi, H. Yousefi Nejad / IJIM Vol. 2, No. 1 (2010) 43-46
P
operator and de ned by SPF (f ) = i2I hf; fi ifi which leads us to reconstruction formula P f = i2I hf; fiiSF 1 fi = i2I hf; SF 1fiifi for all f 2 H and also we have AId SF BID for frame operator. Let fWi g be a family of closed subspaces of H and let fvi g be a family of weights, i.e. vi > 0 for all i 2 I . Then W = f(Wi ; vi)giP 2I is a fusion frame for H if there exist constants 0 < C D < 1 such that C < kf k2 i2I vi2 kwi (f )k2 Dkf k2 for all f 2 H . The numbers C; D are called lower and upper fusion frames bounds, respectively. The family W is called a tight fusion frame if C = D, it is a Parseval fusion frame if C = D = 1 , it is a v-uniform fusion frame if vi = vj = v for all i; j 2 I and an orthonormal fusion basis for H if H = i2I Wi. If the right-handed of mentioned inequality holds, then we say that W is a fusion Bessel sequence and call B the fusion Bessel bound.The operator SF : H ! H P de ned by SF (f ) = i2I hf;Pfi ifi is called the fusion P frame operator which leads us to reconstruction formula f = i2I vi2 SW1 Wi (f ) = i2I vi2 Wi SW1 (f ) for all f 2 H . Also we have CId SW DId for frame operator.
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Main Results
In the following Lemma [1] Casazza and Kutyniok have proved if W = f(Wi ; vi )gi2I is a fusion frame for H , then the intersection of a closed subspace V of H with the family of subspaces fWi gi2I of H which have the same weights, i.e. Wv = f(Wi \ V; vi )gi2I ,is a fusion frame .
Lemma 2.1. Let V be a subspace of H and W = fWi ; vi gi2I be a fusion frame forH with bounds C; D then WV = f(Wi \ V; vi )gi2I is a fusion frame for V with bounds C; D Remark 2.1. The authors have used the following equation in proof of the above Lemma
X v2k i2I
i
2=
Wi (f )k
X v 2 k i2I
i
2
Wi \V (f )k
(for all f 2 H )
(2.1)
But this equation is not correct in general. Actually, the right-hand side of Eq. (2.1) could be equal toPzero. In fact, if Wi P \ V = fog; (8i 2 I ) then Wi\V (f ) = 0 for all f 2 H . 2 2 Therefore i2I vi kWi (f )k = i2I vi2 kWi \V (f )k2 = 0. Thus C kf k2 o which is in contradiction with o < C D < 1
Now, we state a counter example as follow. Example 2.1. Let n 2 N and n 2, set N = f1; 2; :::; ng N . Then E = fei gi2N is a canonical orthonormal basis for l2 (N ), where ej = fij gi2N ; (8j 2 N ). Let W1 = spanfe1 + e2 g and for every i 2 N; i 6= 1; Wi = spanfei g. Then fWi gi2N is a family of subspaces of l2 (N ). Put V = spanfe1 g, we show that there exists a family of weights fvi gi2I such that W = f(Wi ; vi)gi2I . Let ffi gi2I be an orthonormal basis for a subspace W of H then X X X W f = hW f; fiifi = hf; W fiifi = hf; fiifi (8f 2 H ) so
i2I
i2I
i2I
w1 f = hf; e1p+2e2 i e1p+2e2 = 12 hf; e1 + e2 i(e1 + e2 ) = 12 hf; e1 ie1 + 12 hf; e1 ie2 + 12 hf; e2 ie1 + 21 hf; e2 ie2 44
H. R. Rahimi, H. Yousefi Nejad / IJIM Vol. 2, No. 1 (2010) 43-46
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we note Wi f = hf; ei iei ; (8i 2 N; i 6= 1; 8f 2 H ), so kWi f k2 = jhf; ei ij2 . On the other hand 1 1 1 1 2
kW f k = h 2 hf; e1 ie1 + 2 hf; e1 ie2 + 2 hf; e2 ie1 + 2 hf; e2ie2 ; 1 hf; e1 ie1 + 1 hf; e1 ie2 + 1 hf; e2 ie1 + 1 hf; e2 ie2 i 2 2 2 2 = 12 jhf; e1 ij2 + 21 jhf; e2 ij2 + jhf; e1 ijjhf; e2 ij p let v1 = 2; vi = 1 where,i = 6 1; i 2 N we show that W = f(Wi ; vi )gi=I is a fusion frame of subspaces for l2 (N ). We have Pni=1 vi2kWi (f )k2 = 2[ 1 jhf; e1ij2 + 1 jhf; e2ij2 + jhf; e1ij jf; e2ij] + Pni=2 jhf; eiij2 2 P2 = ni=1 jhf; ei ij2 + jhf; e2 ij2 + 2jhf; e1 ij jhf; e2 ij = kf k2 + jhf; e2 ij2 + 2jhf; e1ij jhf; e2 ij If jhf; e1 ij jhf; e2 ij, then 2jhf; e2 ij2 2jhf; e1 ij jhf; e2 ij 2jhf; e1 ij2 Thus
kf k2 kf k2 + 3jhf; e2 ij2 If j hf; e1 ij
