A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES. S. K. Kaushik, Ghanshyam Singh and Virender University of Delhi and M.L.S.

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1 GLASNIK MATEMATIČKI Vol. 43(63)(2008), A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES S. K. Kaushik, Ghashyam Sigh ad Vireder Uiversity of Delhi ad M.L.S. Uiversity, Idia Abstract. Block sequeces with respect to frames i Hilbert spaces have bee defied. Examples have bee provided to show that a block sequece with respect to a give frame may ot eve be a Bessel sequece. Also, a ecessary ad sufficiet coditio uder which a block sequece with respect to a frame is a frame has bee give. Further, applicatios of block sequeces to obtai Fusio frames ad Fusio frame systems have bee give. Fially, a problem has bee posed ad observed that a affirmative aswer to this problem gives a affirmative aswer to the Feichtiger Cojecture. 1. Itroductio I 1952, Duffi ad Schaeffer [13] itroduced frames for Hilbert spaces. It took more tha 30 years to realize the importace of frames. But, after the ladmark paper of Daubechies, Grossma ad Meyer [12], i 1986, the theory of frame bega to be more widely studied. For a itroductio to frames, oe may refer to [3, 10, 11, 14, 15, 16]. Casazza [4], ad Beedetto ad Fickus [2] have studied frames i fiite dimesioal spaces which attracted more attetio due to their use i sigal ad image processig. For sigal recostructio without phase iformatio, oe may refer to [1]. Frames, ow a days, are mai tools for use i sigal ad image processig, data compressio, samplig theory, optics, filter baks, sigal detectio, time frequecy aalysis etc. A umber of ew applicatios have emerged which caot be modeled aturally by oe sigle frame system. I such cases, the data assiged to oe sigle frame system becomes too large to be hadled umerically. So, it would be beeficial to split large frame system ito a set of much smaller systems 2000 Mathematics Subject Classificatio. 42C15, 42A38. Key words ad phrases. Frame, Bessel sequece, block sequece. 387

2 388 S. K. KAUSHIK, G. SINGH AND VIRENDER ad to process the data locally withi each subsystem effectively. Thus, a distributed frame theory for a set of local frame systems is required. I this directio, a theory based o Fusio frames was developed i [6, 8, 9, 18] which provides a framework to deal with these applicatios ad to derive efficiet ad robust algorithms. Aother well kow requiremet i the frame theory is to decompose a bouded frame ito fiite uio of Riesz basic sequeces. This is kow as Feichtiger cojecture ad is coected to the famous Kadiso-Siger cojecture [17]. It was show i [5] that Kadiso-Siger cojecture implies Feichtiger cojecture. For more results regardig the Feichtiger cojecture, oe may refer to [5, 7]. I the preset paper, we defie block sequeces is Hilbert spaces ad give examples to show that a block sequece with respect to a give frame eed ot be a frame (frame sequece). Also, a ecessary ad sufficiet coditio uder which a block sequece with respect to a give frame is a frame has bee give. Further, we discuss applicatios of block sequeces to obtai fusio frames ad fusio frame systems. Fially, we pose a problem ad observe that a affirmative aswer to this problem gives a affirmative aswer to the Feichtiger cojecture. 2. Prelimiaries Throughout the paper, H will deote a ifiite dimesioal Hilbert space, { k } a ifiite icreasig sequece i N, [x ] the closed liear spa of {x }, for ay set D, O(D) will deote the cardiality of D. Defiitio 2.1. A sequece {x } H is called a frame for H if there exist costats A, B > 0 such that (2.1) A x 2 N x, x 2 B x 2, x H. The positive costats A ad B, respectively, are called lower ad upper frame bouds of the frame {x }. The iequality (2.1) is called the frame iequality. The frame {x } H is called tight if it is possible to choose A, B satisfyig iequality (2.1) with A = B ad is called ormalized tight if A = B = 1. The frame {x } H is called exact if removal of ay arbitrary x reders the collectio {x } o loger a frame for H. A sequece {x } H is called a Bessel sequece if it satisfies upper frame iequality i (2.1). A sequece {x } H is said to be a frame sequece for H if {x } is a frame for [x ].

3 A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES Mai Results Defiitio 3.1. A sequece {y } i a Hilbert space H is said to be a block sequece with respect to a sequece {x } i H if it is of the form (3.1) y = α i x i 0, N, i D where D s are fiite subsets of N with D D m = φ, m, ad α i s are ay scalars. N D = N The followig observatios arise aturally i wake of the block sequeces with respect to frames. Observatios. (I) A block sequece with respect to a frame i a Hilbert space may or may ot be a frame for H. Ideed: (a) Let {x } be ay frame for H, the for scalars α i = 1, i N ad D = {}, N, {y } (as give i (3.1)) is a frame for H. (b) Let {x } be a sequece of orthoormal uit vectors i H. The {x } is a frame for H. Take α i = 1, i N ad D 1 = {1, 2, 3}; D = {2 + }, for all 2. The the block sequece {y } is a frame sequece for H but it is ot a frame for H. (II) A block sequece with respect to a frame i H may ot eve be a frame sequece for H. Ideed, let {e } be the sequece of orthoormal uit vectors i H ad {x } i H be defied by x 1 = e 1 ; x 2 = x 2+1 = e +1, for all 1. The {x } is a frame for H. Take α i = 1, i N ad D 1 = {1}; D = {2 2, 2 1}, for all 2. The the block sequece {y } is ot a frame sequece for H. However, [y ] = H. (III) A block sequece with respect to a sequece i H which is ot eve a frame for H may be a frame for H. Ideed, let {e } be a sequece of orthoormal uit vectors i H. Defie {x } i H by x = e, N. The {x } is ot a frame for H. Take α =, N ad D = {}, N. The the block sequece {y } with respect to {x } is a frame for H. (IV) A block sequece {y } with respect to a frame {x } i H with if y > δ, for all N, for some δ > 0, may fail to esure that {y } is a frame sequece for H (see Observatio (II)). (V) A block sequece {y } with respect to a frame {x } with [y ] = [x ] may also fail to esure that {y } is a frame sequece for H. Ideed, let {x } be the sequece of orthoormal uit vectors i H. Take α i = 1 i,

4 390 S. K. KAUSHIK, G. SINGH AND VIRENDER i N ad D = {}, N. The [y ] = [x ]. But {y } is ot a frame sequece for H. I view of the above observatios, oe may ask for sufficiet coditios uder which a block sequece with respect to a frame is a Bessel sequece (frame) for H. We prove the followig result i this directio. Theorem 3.2. Let {x } be a frame for H ad {y } be a block sequece with respect to {x }. The {y } is a frame for H if α i <, sup{o(d ) : N} < ad if α i 2 i D i,j D α i ᾱ j > 0. Further, if {x } is exact, the {y } is also exact. Proof. Let 0 < A B < be costats such that sup 1 i< A x 2 x, x 2 B x 2, x H. Now y, x 2 = α i x i, x i D sup α i 2 1 i< 2 K 0 ( i D x, x 2 (where K 0 = sup{k : k = O(D ), cardiality of D }) K 0 sup 1 i< α i 2 B x 2, x H. Therefore, {x } is a Bessel sequece for H with boud K 0 Agai y, x 2 ), sup 1 i< α i 2 B = B 0. = 2 α i x i, x i D = α i 2 x i, x 2 + α i ᾱ j x i, x x, x j i D i,j D α i 2 x i, x 2 x, x 2 α i ᾱ j. i D i D i,j D

5 A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES 391 Therefore where A 0 = aa ad a = if y, x 2 A 0 x 2, i D α i 2 i,j D x E α i ᾱ j. Hece {y } is a frame for H with bouds A 0 ad B 0. Further, if {y } is ot exact, the, for some m N y m = j m β j y j, where β j 0 for some j m. This gives α i x i = β j α k x k, i D m j m k D j where α i0 0 for some i 0 D m. Therefore, for some c 0, i 0, we may write x i0 = =i 0 c x. Hece {x } is ot exact. Remark 3.3. (i) The coditio sup{o(d ) : N} < ca ot be dropped { as if we cosider H = l 2 with} orthoormal basis {e } ad let 1 D = ( 1) + k : k = 1, 2,...,, N. Clearly D D m = φ 2 for m ad D = N. Also ote that sup{o(d ) : N}. N Let {y } be a block sequece defied by the relatio y = i D α i e i, where α i = 1 for all i N. The, for x = 3/2 e i i H, x < 2, but x, y 2 does i D ot coverge ad hece {y } is ot a Bessel sequece, so ca ot be a frame for H. (ii) I Theorem 3.2, the coditio that if α i ᾱ j > 0 i D α i 2 i,j D is ot ecessary (see the details of Observatio I(b)). Also, we ca ot drop this coditio as the block sequece {y } i Observatio (V) is

6 392 S. K. KAUSHIK, G. SINGH AND VIRENDER ot a frame ad i this case if α i 2 i D i,j D α i ᾱ j 0 (iii) The coditio that sup 1 < α < is ot ecessary. Ideed, let {e } is a sequece of orthoormal uit vectors ad {x } is defied as x 2 1 = e ad x 2 = e, N. The the block sequece {y } defied by y = x x 2, N is also a frame for H, but sup α i. We ow give a ecessary ad sufficiet coditio uder which a block sequece with respect to a frame is a frame. Theorem 3.4. Let {x } be a frame for H ad {y } be a block sequece with respect to {x }. Let T : l 2 (N) l 2 (N) be a bouded liear operator such that T({ x, x }) = { y, x }, x H. The {y } is also a frame for H if ad oly if there exists a λ > 0 such that { y, x } λ { x, x }, x H. Proof. Let 0 < A B < be such that (3.2) A x 2 x, x 2 B x 2, x H. The y, x 2 = { y, x } 2 λ 2 A x 2, x H. Also y, x 2 = { y, x } 2 = T({ x, x }) 2 T 2 B x 2, x H. Hece {y } is frame for H with bouds λ 2 A ad T 2 B. Coversely, let {y } is a frame for H with bouds A y ad B y. The, for ay x H, we have A y x 2 y, x 2 B y x 2, x H. Therefore, by (3.2) A y x, x 2 A y x 2 B y, x 2, x H.

7 A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES 393 Put A y B = λ2. The λ > 0 such that { y, x } λ { x, x }, x H. Applicatios Let {x } be a frame for H ad let V = [x i ] i D, N. The {V } is a sequece of subspaces of H such that V = H. Therefore, each x H ca be expressed as (3.3) x = y i, i=1 where y i V i, i N. The represetatio i (3.3) may ot be uique. Note that, for each N, {x i } i D is a frame for the subspace V. Defie v : H V by v (x) = α i x i V, N, x H. i D The oe ca fid costats 0 < A B < such that A x 2 v i (x) 2 B x 2, x H. i=1 Therefore (V, v ) N is a Fusio frame for H ad (V, v, {x i } i D ) is a Fusio frame system for H (Fusio frames ad Fusio frame systems were itroduced ad studied recetly by Casazza ad Kutyiok ad others i [6, 8, 9, 18]). If {x } is exact frame for H, the there exists a sequece of fiite subsets {D } of N (e.g. take D = {}, N) with D = N such that H = N V, where V = [x i ] i D. N Also, {f } be the sequece of uit vectors i H. Defie a sequece {g } i H by g = 1 f, N. Let k = k 1 + (k 1), k N ad 0 = 1. The { k } is a ifiite icreasig sequece i N. Now defie {h } i H by h 1 = g 1, h k = h k +1 = h k +2 = = h k+1 1 = g k, k 2. The {h } is a tight o-exact frame for H. Takig D k = { k, k + 1,..., k+1 1}, k N ad V k = [h i ] i Dk, k N, we get H = N V. I view of the above discussio, it is atural to raise the followig problem.

8 394 S. K. KAUSHIK, G. SINGH AND VIRENDER Problem 3.1. Is it always possible to have a sequece of fiite subsets {D } of N with D = N such that H = V, where V = [x i ] i D ad N {x } is a bouded frame for H? Remark 3.5. A affirmative aswer to this problem solves the Feichtiger cojecture i affirmative. Ideed, let {D } N be a sequece of fiite subsets of N with D = N such that H = V, where V = [x i ] i D. N N Defie G i = {x } Di ad for each j N, choose {y j i } i N such that y j i = jth elemet of G i. The for each j, {y j i } is a Riesz basic sequece for H ad {x } = {y j i }. j Ackowledgemets. The authors thak the referees for thoroughly readig the paper ad for providig valuable commets ad useful suggestios for the improvemet of the paper. The research of first author is partially supported by the UGC (Idia) (Letter No.F.6-1(52)/2007 (MRP/Sc/NRCB)) ad the research of third author is supported by the CSIR (Idia) (vide letter No. 09/172(0053)/2006- EMR-I dated ). Refereces [1] R. Bala, P. G. Casazza ad D. Edidi, O sigal recostructio without phase, Appl. Comput. Harmo. Aal. 20 (2006), [2] J. J. Beedetto ad M. Fickus, Fiite ormalized tight frames, Frames, Adv. Comput. Math. 18 (2003), [3] P. G. Casazza, The art of frame theory, Taiwaese J. Math. 4 (2000), [4] P. G. Casazza, Custom buildig fiite frames, i: Wavelets, frames ad operator theory, 61-86, Cotemp. Math. 345, Amer. Math. Soc., Providece, RI, [5] P. G. Casazza, O. Chirstese, A. Lider ad R. Vershyi, Frames ad Feichtiger Cojecture, Proc. Amer. Math. Soc. 133 (2005), [6] P. G. Casazza ad G. Kutyiok, Robustess of fusio frames uder erasures of subspaces ad of local frame vectors, Radom trasforms, geometry, ad wavelets (New Orleas, LA, 2006), Cotemp. Math., Amer. Math. Soc., Providece, RI (to appear). [7] P. G. Casazza, G. Kutyiok, D. Speegle ad J. C. Tremai, A decompositio theorem for frames ad the Feichtiger cojucture, Proc. Amer. Math. Soc. 136 (2008), [8] P. G. Casazza, G. Kutyiok ad S. Li, Fusio frames ad distributed processig, Appl. Comput. Harmo. Aal. 25 (2008), [9] P. G. Casazza, G. Kutyiok, S. Li ad C. J. Rozell, Modelig sesor etworks with fusio frames. Wavelets XII (Sa Diego, CA, 2007), 67011M M-11, SPIE Proc. 6701, SPIE, Belligham, WA, [10] O. Christese, A Itroductio to Frames ad Riesz Bases, Birkhäuser Bosto, Ic., Bosto, MA, [11] O. Christese, Liear combiatios of frames ad frame packets, Z. Aal. Aweduge 20 (2001), [12] I. Daubechies, A. Grossma ad Y. Meyer, Pailess o-orthogoal expasios, J. Math. Phys. 27 (1986),

9 A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES 395 [13] R. J. Duffi ad A. C. Schaeffer, A class of oharmoic Fourier series, Tras. Amer. Math. Soc. 72 (1952), [14] S. J. Favier ad R. A. Zalik, O the stability of frames ad Riesz bases, Appl. Comput. Harmo. Aal. 2 (1995), [15] D. Gabor, Theory of commuicatios, J. IEE (Lodo) 93 (1946), [16] C. Heil ad D. Walut, Cotiuous ad discrete wavelet trasform, SIAM Rev. 31 (1989), [17] R. V. Kadiso ad I. M. Siger, Extesios of pure states, Amer. J. Math. 81 (1959), [18] G. Kutyiok, A. Pezeshki, A. R. Calderbak ad T. Liu, Robust dimesio reductio, fusio frames, ad Grassmaia packigs, preprit. S. K. Kaushik Departmet of Mathematics Kirori Mal College, Uiversity of Delhi Delhi Idia shikk2003@yahoo.co.i G. Sigh Departmet of Mathematics ad Statistics, Uiversity College of Sciece, M.L.S. Uiversity, Udaipur (Rajastha) Idia Ghashyamsrathore@yahoo.co.i Vireder Departmet of Mathematics ad Statistics, Uiversity College of Sciece, M.L.S. Uiversity, Udaipur (Rajastha) Idia vireder57@yahoo.com Received: Revised: &