SUMS OF HILBERT SPACE FRAMES PETER G. CASAZZA, SOFIAN OBEIDAT, SALTI SAMARAH, AND JANET C. TREMAIN

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1 SUMS OF HILBET SPACE FAMES PETE G. CASAZZA, SOFIAN OBEIDAT, SALTI SAMAAH, AND JANET C. TEMAIN Abstract. We give simple ecessary ad sufficiet coditios o Bessel sequeces {f i } ad {g i } ad operators L 1, L 2 o a Hilbert space H so that {L 1 f i + L 2 g i } is a frame for H. This allows us to costruct a large umber of ew Hilbert space frames from existig frames. 1. Itroductio Frames for Hilbert spaces were itroduced by Duffi ad Schaeffer [9] as a part of their research i o-harmoic Fourier series. Their work o frames was somewhat forgotte util 1986 whe Daubechies, Grossma ad Meyer [14] brought it all back to life durig their fudametal work o wavelets. Today, frame theory plays a importat role ot just i sigal processig, but also i dozes of applied areas (See [2, 3]). Holub [13] showed that if {x } is ay ormalized basis for a Hilbert space H ad {f } is the associated dual basis of coefficiet fuctioals, the the sequece {x + f } is agai a basis for H. I this paper we study cases i which ew frames ca be obtaied from old oes. Throughout H deotes a separable Hilbert space. A frame for H is a family of vectors f i H, for i I for which there exist costats A, B > 0 satisfyig: (1.1) A f 2 i I f, f i 2 B f 2 for all f H. A ad B are called the lower ad upper frame bouds respectively. If A = B, this is called a A-tight frame. Ad if A = B = 1, it is a Parseval frame. If we have just the upper iequality, we call {f i } a B-Bessel sequece. If {f i } i I is a B-Bessel sequece, we defie its aalysis operator as T : H l 2 (I) by: T (f) = { f, f i } i I. The first ad fourth authors were supported by NSF DMS

2 2 P.G. CASAZZA, S. OBEIDAT, S. SAMAAH, AND J.C. TEMAIN The adjoit of the aalysis operator is the sythesis operator give by: T ({a i } i I ) = i I a i f i. If the aalysis operator is bouded, we defie the frame operator S = T T ad ote that S is a positive, self-adjoit operator which is ivertible o H if ad oly if {f i } is a frame for H. If {f i } is a frame, the every f H has a represetatio of the form (1.2) f = i I f, S 1 f i f i = i I f, f i S 1 f i = i I f, S 1/2 f i S 1/2 f i., For a itroductio to frame theory we recommed [4, 7]. For a itroductio to Gabor frames we recommed Gröcheig [10]. 2. Begiigs We wat to observe that if we have ay frame {f i } i I for a Hilbert space H with frame bouds A, B ad frame operator S, the for all real umbers a, {f i + S a f i } i I is also a frame for H with frame operator (I + S a ) 2 S ad frame bouds I + S a 2 A, I + S a 2 B. I particular, {f i + Sf i }, {f i + S 1 f i } (i.e. The frame added to its cooical dual frame) ad {f i + S 1/2 f i } (i.e. The frame added to its caoical Parseval frame) are all frames for H. We start with a well kow result. Dager: There is a problem with the ext theorem. See [1]. Ad this leads to problems with Corollaries 2.2 ad 2.3. Propositio 2.1. Let {f i } i I be a frame for H with frame operator S, frame bouds A B ad let T : H H be a bouded operator. The {T f i } i I is a frame for H if ad oly if T is ivertible. Moreover, i this case the frame operator for {T f i } is T ST ad the ew frame bouds are T 1 2 A, T 2 B. Proof. For ay f H we have ( ) f, T fi T f i = T T f, f i f i = T ST f. Tryig to add a frame {f i } to Lf i } ca be problematic i geeral sice we could have Lf i = f i. However, the followig corollary of Propositio 2.1 shows that this is all that ca really go wrog.

3 SUMS OF HILBET SPACE FAMES 3 Corollary 2.2. If {f i } i I is a frame for H ad L : H H is a bouded operator, the {f i + Lf i } is a frame for H if ad oly if I + L is ivertible. I this case, the frame operator for the ew frame is (I + L)S(I + L ) ad the frame bouds are I + L 2 A, I + L 2 B. I particular, if L is a positive operator (or just L > 1) the {f i + Lf i } is a frame with frame operator S + LS + SL + LSL. The above corollary shows that all of our earlier sums give ew frames for H. Corollary 2.3. If {f i } i I is a frame for H ad P is a orthogoal projectio o H, the for all a 1 we have that {f i + ap f i } i I is a frame for H. Proof. We just write: ad apply Corollary 2.2. f i + ap f i = (1 + a)f i a(i P )f i, The reaso we wat to add frames together is to produce frames with better properties for particular applicatios. Let us look at a simple example. ecall that a frame is ɛ-early Parseval if its frame bouds A, B satisfy: 1 ɛ A B 1 + ɛ. Example 2.4. Let {f m } M m=1 be a ɛ-early Parseval frame for H N with frame bouds A, B ad frame operator S. Let g m = 1 2 (f m + S 1 f m ), for all m = 1, 2,, M. The {g m } M m=1 is a frame with frame bouds 1, 1 + ɛ 4 which is close to {f m}. Proof. The frame operator for the frame {g m } is [ ] 2 1 S 0 = 2 (I + S 1 ) S = 1 2 I + S + S 1. 4 Let {e } N =1 be a eigebasis for S with respective eigevalues {λ } N =1. The {e } N =1 is a eigebasis for S 0 with respective eigevectors Sice f m } is ɛ-early Parseval, Hece, (λ + λ 1 ). λ + 1 λ max{1 + ɛ ɛ, 1 ɛ ɛ } 2 + ɛ (λ + λ 1 ) ɛ 4.

4 4 P.G. CASAZZA, S. OBEIDAT, S. SAMAAH, AND J.C. TEMAIN Fially, we check how close the ew frame is to the old frame. M M f m g m 2 = 1 2 (f m S 1 f m ) 2 m=1 = = = = m=1 M 1 2 (I S 1 )f m 2 m=1 M m=1 =1 N 1 2 (1 1 ) 2 f m, e 2 λ N 1 2 (1 1 M ) 2 f m e 2 λ =1 2 1 ɛ 2 1 ɛ (1 + ɛ) 2 ( ɛ ) ( ) ɛ. 2 1 ɛ The reaso the above frame is iterestig is that it is much closer to {f i } tha the earest Parseval frame which is {S 1/2 f i } [5] ad its frame bouds are much better tha the origial. m=1 3. Sums of Bessel Sequeces Now we wat to show that a frame ca be added to ay of its alterate dual frames to yield a ew frame. ecall, if {f i } i I is a frame, the caoical dual frame is {S 1 f i } ad satisfies the property that for all f H, f = i I f, f i S 1 f i. A frame {g i } is called a alterate dual frame if for all f H, f = f, f i g i. i I We start by extedig our earlier ideas. Propositio 3.1. Let {f i } i I ad {g i } i I be Bessel sequeces i H with aalysis operators T 1, T 2 ad frame operators S 1, S 2 respectively. Also let L 1, L 2 : H H. The followig are equivalet: (1) {L 1 f i + L 2 g i } i I is a frame for H. (2) T 1 L 1 + T 2 L 2 is a ivertible operator o its rage. (3) We have S = L 1 S 1 L 1 + L 2 S 2 L 2 + L 1 T 1 T 2 L 2 + L 2 T 2 T 1 L 1 > 0. Moreover, i this case, S is the frame operator for {L 1 f i + L 2 g i } i I.

5 SUMS OF HILBET SPACE FAMES 5 Proof. (1) (2): {L 1 f i +L 2 g i } i I is a frame if ad oly if its aalysis operator T is ivertible o its rage where T f = { f, L 1 f i + L 2 g i } = { L 1f, f i + L 2f, g i } = T 1 L 1f + T 2 L 2f. (2) (3): The frame operator for our family is S = (T 1 L 1 + T 2 L 2) (T 1 L 1 + T 2 L 2) = L 1 S 1 L 1 + L 2 S 2 L 2 + L 1 T 1 T 2 L 2 + L 2 T 2 T 1 L 1. Our family of vectors is a frame if ad oly if S > 0. The followig theorem eables oe to get a frame from a combiatio of a kow frame ad a Bessel sequece. Theorem 3.2. Let {f i } i I be a frame for a Hilbert space H with frame operator S 1 ad let {g i } i I be a Bessel sequece i H with frame operator S 2. If (f) = i I f, g i f i, is a positive operator, the {f i + g i } i I is a frame for H with frame operator S S 2. Proof. Let T 1, T 2 be the aalysis operators for {f i }, {g i } respectively. Lettig L 1 = I = L 2 i Propositio 3.1 we see that the frame operator for {f i + g i } i I is S 0 = S 1 + S 2 + T 1 T 2 + T 2 T 1 = S 1 + S As a applicatio of the theorem we have Corollary 3.3. If {f i } is a frame for H with frame operator S ad {g i } is a alterate dual frame the {S a f i + S b g i } is a frame for H for all real umbers a, b. Proof. We let L(f) = f, S b g i S a f i = S a+b (f). i I That is, L 0. So {S a f i + S b g i } i I is a frame by Theorem 3.2. We do ot ecessarily eed {g i } to be a alterate dual frame above. Theorem 3.4. Let {f i } be a frame for H with frame operator S that is ormbouded below. If {g i } H such that f = i I f, g i f i ucoditioally for all f H the {S a (f i ) + S b g i } is a frame for H for all real umbers a, b.

6 6 P.G. CASAZZA, S. OBEIDAT, S. SAMAAH, AND J.C. TEMAIN Proof. Note that the assumptio is that = I 0. Also, sice {f i } i I is bouded, assumig f, g i f i, coverges ucoditioally implies f, g i 2 <, for all f H. i I i I By the Uiform Boudedess Priciple, we have that {g i } i I is a Bessel sequece. So we ca apply Theorem 3.2 to coclude that {f i + g i } i I is a frame for H. Also, L(f) = i I f, S b g i S a f i = S a+b (f). That is, L 0. So {S a f i + S b g i } i I is a frame by Theorem 3.2. The assumptio that the frame is orm bouded below i Theorem 3.4 is ecessary. For example, if {e i } is a orthoormal basis for H let f 2i+1 = e i, f 2i = 1 i e i, g 2i = ie i, g 2i+1 = 0. The for all f H, f, gi f i = f, but {f i + g i } is ot a frame sice it is ot Bessel. Also, the assumptio that the covergece is ucoditioal i Theorem 3.4 is ecessary. For example, let {h i, h i } i I be a Schauder basis for H which is Bessel but ot a frame. Let {e i } i I1 be a orthoormal basis for H. Let The for all f H, {f i } = {e i } i I {h i } i I1, {g i } = {0} i I {h i } i I1. f, e i g i + f, h i g i = 0 + f. i I 1 i I But {f i + g i } is ot a frame sice it is ot Bessel. We ca more carefully do local additio for our frames. Propositio 3.5. Let {f i } i I be a frame for H with frame operator S. Let {I 1, I 2 } be a partitio of I ad let S j be the frame operator for the Bessel sequeces {f i } i Ij, j = 1, 2. The {f i + S a f i } i I1 {f i + S b 2} i I2, is a frame for H for all real umbers a, b.

7 SUMS OF HILBET SPACE FAMES 7 Proof. The frame operator for {f i + S a f i } i I1 (I + S1)S a 1 (I + S1) a = S 1 + 2S1 1+a + S1 1+2a S 1. Similarly for {f i + S b f i } i I2. Hece, the frame operator S 0 for our family satisfies: S 0 S 1 + S 2 = S > 0. is 4. Sums of Gabor Frames For x, y defie the operators E x ad T y o L 2 () by: E x f(t) = e 2πixt, T y f(t) = f(t y). Let g L 2 () ad 0 < ab 1. The (g, a, b) deotes the family: {E mb T a g} m, Z. If this family forms a frame for L 2 () we call it a Gabor frame with widow fuctio g. It is exceptioally difficult to add widow fuctios for Gabor frames to build a ew Gabor frame. Our earlier results work i part because the frame operator S for the Gabor frame (g, a, b) commutes with the operators E mb, T a. So, for example, (g+s c g, a, b) is always aother Gabor frame. But eve simple cases ca become quite complicated i this settig. For example, just lettig g = χ [0,1], h = χ [1,2], it is easily checked that (g, 1, 1) ad (h, 1, 1) are frames (actually orthoormal bases) for L 2 () while (g + h, 1, 1) ad (g + ih, 1, 1) are ot frames [10]. Now let g = χ [0,1/2] + iχ [1/2,1]. The (g, 1, 1) is a Gabor frame while (e g, 1, 1) ad Im g, 1, 1) do ot form frames. Eve if g, h are real valued ad form Gabor frames it is possible that (g + ih, a, b) does ot form a Gabor frame. For example, if g 0 ad (g, a, b) is a Gabor frame the for x 0 (T x g, a, b) certaily forms a Gabor frame. However, (g + T x g, a, b) caot yield a Gabor frame as the followig result shows. Propositio 4.1. For ay g, ad c = 1, ay x ad 0 y, (g + ce y T x g, a, b) does ot form a frame. Proof. Sice {E mb T a (g + ce x T y g)} = {(I + ct x E x+y )(E mb T a g)}. So it suffices to observe that (I + ct x E x+y ) is ot a ivertible operator o L 2 (). To see this let, f = ( 1) k χ [kx,(k+1)x) c k E(x+y). k k=1

8 8 P.G. CASAZZA, S. OBEIDAT, S. SAMAAH, AND J.C. TEMAIN The f 2 = x, while (I + at x E (x+y) f 2 = 2x. Now we will see a case where we ca produce a frame by summig Gabor frames. To simplify the proof, we first recall a few stadard calculatios i this area. The first comes from Walut s PhD thesis (See [12]). Propositio 4.2. If (g, a, b) is a Gabor frame the for all f L 2 () we have: f, E mb T a g 2 = b 1 f(t) 2 g(t a) 2 dt+ m, b 1 f(t)f(t k/b) g(t a)g(t a k/b)dt. The ext calculatio is due to Casazza ad Christese [6]. This is ot exactly what they proved i their theorem. However, their proof works lie for lie i this case. Propositio 4.3. If (g, a, b) is a Gabor frame for L 2 () ad for k Z we let the G k (t) = Z[T a g 2 (t)t a+k/b g 1 (t) T a g 1 (t)t a+k/b g 2 (t)], f(t)f(t k/b)g k (t) dt f(t) 2 G k (t) dt. Now we are ready to prove the mai result cocerig summig Gabor frames. Theorem 4.4. Let (g j, a, b), j = 1, 2 be Gabor frames with frame bouds A j B j respectively ad the fuctios g 1, g 2 are real valued. Assume 1 g 2 (t a)g 1 (t a k/b) g 1 (t a)g 2 (t a k/b) (1 ɛ)(a 1 +A 2 ), b for some 0 < ɛ < 1. The (g 1 + ig 2, a, b) is a Gabor frame. Proof. Applyig Propositios 4.2 ad 4.3 at the appropriate poit we ca calculate: f, E mb T a (g 1 + ig 2 ) 2 = b 1 f(t) 2 (g 1 + ig 2 )(t a) 2 dt+ m, b 1 f(t)f(t k/b) (g 1 + ig 2 )(t a)(g 1 + ig 2 )(t a k/b)dt = b 1 f(t) 2 g 1 (t a) 2 dt + b 1 f(t) 2 g 2 (t a) 2 dt+

9 SUMS OF HILBET SPACE FAMES 9 b 1 f(t)f(f k/b)g 1 (t a)g 1 (t a k/b)dt+ b 1 f(t)f(t k/b)g 2 (t a)g 2 (t a k/b)dt+ b 1 i f(t)f(t k/b)g k (t)dt = f, E mb T a g f, E mb T a g m, m, b 1 i f(t)f(t k/b)g k (t)dt A 1 f 2 + A 2 f 2 b 1 f(t)f(t k/b) G k (t) dt (A 1 + A 2 ) f 2 b 1 (1 ɛ)b(a 1 + A 2 ) f 2 ɛ(a 1 + A 2 ) f 2. efereces [1] A. Najati, M.. Abdollarhpour, E. Osgooei, ad M.M. Saem, More sums of Hilbert space frames, arxiv v1. [2]. Bala, P.G. Casazza, C. Heil ad Z. Ladau, Desity, overcompleteess, ad localizatio of frames. I. Theory, Jour. Fourier Aal. ad Appls. 12 No. 2 (2006) [3]. Bala, P.G. Casazza, C. Heil ad Z. Ladau, Desity, overcompleteess, ad localizatio of frames. II. Gabor Frames, Jour. Fourier Aal. ad Appls. 12 (2006) [4] P.G. Casazza, The art of frame theory, Taiwaese Jour. of Math, 4 No. 2 (2000) [5] P.G. Casazza, Custom buildig fiite frames, Cotemp. Math 345 (2004) [6] P.G. Casazza ad O. Christese, Weyl-Heiseberg frames for subspaces of L 2 (), Proc. AMS 129 No. 1 (2001) [7] O. Christese, A itroductio to frames ad iesz bases, Birkhäuser, Bosto [8] Joh, B. Coway, A course i Fuctioal Aalysis, secod editio spriger-verlag. Newyork Ic [9].J Duffi ad A.C. Schaeffer, A class of oharmoic Fourier series, Tras. Amer. math Soc. 72 (1952), [10] K. H. Gröcheig, Foudatios of time-frequecy aalysis, Birkhäuser, Bosto, [11] C. Heil, Wier amalgam spaces i geeralized harmoic aalysis ad wavelet theory, Ph.D thesis, Uiversity of Marylad, College Park, MD, [12] C. Heil ad D. Walut, Cotiuous ad discrete wavelet trasforms, Siam eview 31 No. 4 (1989) [13] J., Holub, O a property of bases i a Hilbert spaces, Glasgow Math. J. 46 (2004) [14] I. Daudechies, A. Grossma ad Y. Meyer, Pailess oorthogoal expasios, J. Math. Physics 27 (1986),

10 10 P.G. CASAZZA, S. OBEIDAT, S. SAMAAH, AND J.C. TEMAIN Salti Samarah, Jorda Uiversity of Sciece ad Techology, Jorda- Irbid, Departmet of math. ad Stat., address: Sofia Obeidat, Jorda Uiversity of Sciece ad Techology, Jorda- Irbid, Departmet of math. ad Stat., address: Casazza ad Tremai: Departmet of Mathematics, Uiversity of Missouri, Columbia, MO address: