On the Feichtinger conjecture

Electronic Journal of Linear Algebra, Sep 2017

By Pasc Gavruta, Published on 01/01/13

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On the Feichtinger conjecture

Volume 1081-3810 On the Feichtinger conjecture Pasc Gavruta Follow this and additional works at: https://repository.uwyo.edu/ela Recommended Citation - ON THE FEICHTINGER CONJECTURE∗ PASC GA˘ VRUT¸ A† AMS subject classifications. 46C05, 42C15. 1. Introduction. There are many variations of the Feichtinger Conjecture, all equivalent with the following: Every Bessel sequence of unit vectors in a Hilbert space can be partitioned into finitely many Riesz sequences. For details on the Feichtinger Conjecture and the connection with other problems, see [ 2 ], [ 3 ], [ 4 ], [ 9 ] and [ 10 ], and references in these papers. We denote by H a Hilbert space and F = {fn}n∈N ⊂ H. We say that F is a Bessel sequence if there exists B > 0 so that ∞ X |hx, fni|2 ≤ Bkxk2, n=0 547 We say that F is a Riesz sequence (or Riesz basic sequence) if there are A, B > 0 such that A X |ck|2 ≤ k X ckfkk ≤ B X |ck|2 for any finite sequence (ck). Riesz sequences are particular cases of frames (see [ 5 ]). Let be I ⊂ N. If F is a Bessel sequences in H, then FI = {fn}n∈I is clearly also a Bessel sequence in H. In [ 5 ], O. Christensen, using Schur’s test, give conditions on a sequence {fn}n∞=0 to be a Bessel sequence, that it only involves inner products between the elements {fn}n∞=0: Proposition 1.1. [ 5 ] Let {fn}n∞=0 be a sequence in H and assume that there exists a constant B > 0 such that ∞ X |hfj, fki| ≤ B, ∀ j ∈ N. k=0 Then {fn}n∞=0 is a Bessel sequence with bound B. We call this sequences Bessel-Schur sequences. The intrinsically localized sequences, introduced by K. Gr¨ochenig in [ 8 ], are particular cases of Bessel-Schur sequences. In the same paper, he proves that every localized frame is a finite union of Riesz sequences. Another type of localized sequences was introduced by R. Balan, P.G. Casazza, C. Heil, and Z. Landau in [ 1 ]. They show that the Feichtinger Conjecture is true for l1-self-localized frames which are norm-bounded below. l1-self-localized Bessel sequences are also Bessel-Schur sequences. On the other hand, we recall the following definition: Definition 1.2. [ 4 ] A sequence {fn}n∈I of unit vectors in H is called separated if there exists a constant γ < 1 such that |hfn, fki| ≤ γ for any n, k ∈ N, n 6= k. In [ 4 ], the authors, among others, give the following result: Theorem 1.3. Let H be a Hilbert space and let {fn}n∈I be a Bessel sequence of unit vectors in H. Then {fn}n∈I can be partitioned into finitely many separated Bessel sequences. In the following, we prove that the Bessel-Schur sequences satisfies the Feichtinger Conjecture. Also, we prove that every Bessel sequence of unit vectors in a Hilbert space can be partitioned into finitely many uniformly separated sequences. http://math.technion.ac.il/iic/ela 548 to be a Riesz sequence. suppose that Then, FI is a Riesz sequence. bounded: 2. The results. First, we give a condition for a Bessel sequence of unit vectors Theorem 2.1. Let FI = {fn}n∈I be a Bessel sequence of unit vectors. We Proof. If FI is a Bessel sequence in H, then the following operators are linear and 2 T : l (I) → H, T (ci) = X cifi (synthesis operator), Θ : H → l2(I), Θx = {hx, fii}i∈I (analysis operator). Moreover, Θ is the adjoint of T (see [ 5 ]). For c = (cn)n∈I ∈ l2(I), we have and hence, (ΘT )(c) = hX ckfk, fji = X ckhfk, fji j∈I j∈I i∈I k∈I k∈I (ΘT )(c) − c = X ckhfk, fji k6=j j∈I . By Cauchy-Schwartz inequality, it follows 2 k(ΘT )(c) − ck2 = X X ckhfk, fji 2 j∈I k6=j j∈I k∈I k6=j j∈I k∈I k6=j j∈I k∈I k6=j ≤ X X |ck||hfk, fji|1/2 · |hfk, fji|1/2 ≤ X 2 X |ck| |hfk, fji| ≤ σ X 2 X |ck| |hfk, fji| . X |hfk, fji| k∈I k6=j 2 On the Feichtinger Conjecture 549 Changing the order of summation, we obtain k(ΘT )(c) − ck22 ≤ σ X |ck|2 X |hfk, fji| k∈I j∈I j6=k ≤ σ2kck22, and so, kΘT − Ik ≤ σ < 1. Therefore, ΘT is invertible, thus Θ is surjective. It follows that FI is a RieszFischer sequence. From Theorem 3 in [13, Ch. 4, Sec. 2], we have that there exists A > 0 so that A X |ck|2 ≤ k X ckfkk2 k X ckfkk2 ≤ B X |ck|2 for every finite sequence (ck). Since FI is a Bessel sequence, we have for (ck) finite sequence (see [ 5 ]). So, FI is a Riesz sequence. Theorem 2.2. Every Bessel-Schur sequence of unit vectors is union of finite Riesz sequences. Proof. Let j ∈ N fixed. We have: (2.1) and hence, We denote We have aij = aji ≥ 0 and aii = 0. The relation (2.1) is equivalent with ∞ X |hfj , fii| ≤ B, i=0 X |hfj , fii| ≤ B − 1, for any j ∈ N. i=0 i6=j aij = (|hfj , fii|, 0, j 6= i, j = i. sup X aij ≤ B − 1. j∈N i∈N sup X aij ≤ j∈Ip i∈Ip B − 1 ; 2 By Mills’ Lemma (see [6, Ch. X] or [ 12 ]) there is a partition N = I1 ∪ I2 such that http://math.technion.ac.il/iic/ela 550 By iteration, for any m ≥ 1, there is a partition N = I1 ∪ I2 ∪ . . . ∪ I2m such that j∈Ip i∈Ip sup X aij ≤ B − 1 2m , 2m 3. An equivalent form of the Feichtinger conjecture. We consider the following class of sequences. Definition 3.1. Let FI = {fn}n∈I (...truncated)


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Pasc Gavruta. On the Feichtinger conjecture, Electronic Journal of Linear Algebra, 2018, Volume 26, Issue 1,