Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract. We ivestigate the cosequeces of the lower frame coditio ad the lower Riesz basis coditio without assumig the existece of the correspodig upper bouds. We prove that the lower frame boud is equivalet to a expasio property o a subspace of the uderlyig Hilbert space H, ad that the lower frame coditio aloe is ot eough to obtai series represetatios o all of H. We prove that the lower Riesz basis coditio for a complete sequece implies the lower frame coditio ad ω-idepedece; uder a extra coditio the statemets are equivalet. Keywords: Frames, Riesz-Fischer sequeces, Riesz basis AMS subject classificatio: 4C15 1. Itroductio Let H be a separable Hilbert space. Recall that a sequece {f i } H is a frame if, for some costats A, B > 0, A f f, f i B f, f H. (1.1) The sequece {f i } is a Riesz basis if spa {f i} = H ad there exist costats A, B > 0 such that, for all fiite scalar sequeces {c i }, A c i c i f i B ci. (1.) Peter G. Casazza: Uiv. of Missouri, Dept. Math., Columbia Mo 6511, USA O. Christese: Tech. Uiv. of Demark, Dept. Math., Bdg 303, 800 Lygby, Demark Shidog Li: Sa Fracisco State Uiversity, USA A. M. Lider: Tech. Uiv. of Muich, Dept. Math., D-8090 Muich, Germay pete@casazza.math.missouri.edu, Ole.Christese@mat.dtu.dk shidog@hilbert.sfsu.edu, lider@mathematik.tu-mueche.de P. Casazza was supported by NSF DMS 010686, ad O. Christese was supported by the WAVE-programe, spsored by the Daish Scietific Research Coucil. ISSN 03-064 / $.50 c Helderma Verlag Berli
306 P. Casazza et al. A Riesz basis is a frame; ad if {f i } is a frame, there exists a dual frame {g i } such that f = f, g i f i = f, f i g i, f H. (1.3) I this ote we ivestigate the cosequeces of the lower bouds i (1.1) ad (1.) without assumig the existece of the upper bouds. Note that the lower coditio i (1.1) implies that every f H is uiquely determied by the ier products f, f i (i N): if f, f i = g, f i for all i N, the f = g. That is, i priciple we ca recover every f H based o kowledge of the sequece { f, f i }. We prove that we actually obtai a represetatio of type (1.3) for certai f H. The questio whether the represetatio ca be exteded to work for all f H has bee ope for some time. We preset a example where it ca ot be exteded.. Some defiitios ad basic results For coveiece we will idex all sequeces by the set of atural umbers N. Defiitio.1. Let {f i }, {g i} H. We say that {f i} (i) is a Riesz-Fischer sequece if there exists a costat A > 0 such that A c i c i f i for all fiite scalar sequeces {c i } (ii) satisfies the lower frame coditio if there exists a costat A > 0 such that A f f, f i for all f H (iii) is a Bessel sequece if there exists a costat B > 0 such that f, f i B f for all f H (iv) is miimal if f j / spa {f i } i j for all j N (v) is ω-idepedet if c if i = 0 implies c i = 0 for all i N (vi) is complete if spa {f i } = H (vii) ad {g i } are biorthogoal if f i, g j = δ i,j (Kroecker s δ symbol). For a give family {f i } H, our aalysis is based o the sythesis operator { } T : D(T ) := {c i } l c i f i coverges H, T {c i } = c i f i (.1) ad o the aalysis operator { } U : D(U) := f H f, f i < l, Uf = { f, f i }. (.) The Lemma below is stated i [4: Sectios 1.8 ad 4.].
Riesz-Fischer Sequeces 307 Lemma.. Let {f i } H. The {f i} (i) has a biorthogoal sequece if ad oly if {f i } is miimal; if a biorthogoal sequece exists, it is uique if ad oly if {f i } is complete. (ii) is a Riesz-Fischer sequece if ad oly if the associated aalysis operator is surjective. We collect two other characterizatios of Riesz-Fischer sequeces. Apparetly, they have ot bee stated explicitely before; they ca be proved usig methods developed i [7]. Propositio.3. (i) Let {e i } be a orthoormal basis for H. The Riesz-Fischer sequeces i H are precisely the families {V e i }, where V is a operator o H (havig {e i } i the domai), which has a bouded iverse V 1 : R(V ) H. (ii) The Riesz-Fischer sequeces i H are precisely the families for which a biorthogoal Bessel sequece exists. Example.4. Let {e i } be a orthoormal basis ad cosider {g i} = {e i + e i+1 }. The {g i} is complete ad miimal; it is also a Bessel sequece, but ot a frame. A straightforward calculatio shows that the biorthogoal system is give by { i f i = k=1 ( 1)k e k if i is eve i k=1 ( 1)k+1 e k if i is odd ad {f i } is a Riesz-Fischer sequece by Propositio.3. 3. The lower frame coditio I this sectio we aalyze the relatioship betwee the lower frame coditio ad Riesz-Fischer sequeces. Our results geeralize the kow results because we do ot assume that the sequece is a Bessel sequece. Lemma 3.1. For a arbitrary sequece {f i } H, the associated aalysis operator U is closed. Furthermore, {f i } satisfies the lower frame coditio if ad oly if U has closed rage ad is ijective. Proof. That U is closed follows by a stadard argumet. To prove that {f i } satisfies the lower frame coditio if ad oly if U has closed rage ad is ijective, ote that the existece of a lower frame boud implies ijectivity of U. Sice U is closed, U 1 is closed. Thus, by the closed graph theorem, U has closed rage if ad oly if U 1 is cotiuous o R(U), which is obviously equivalet to the existece of a lower frame boud
308 P. Casazza et al. Recall that a frame is a Riesz basis if ad oly if it is ω-idepedet. The Theorem below geeralizes this result to the case where {f i } satisfies oly the lower frame coditio. It coects the cocepts listed i Defiitio.1: Theorem 3.. Let {f i } H with associated sythesis operator T. Cosider the followig statemets: (i) {f i } is a complete Riesz-Fischer sequece. (ii) {f i } is miimal ad satisfies the lower frame coditio. (iii) {f i } is ω-idepedet ad satisfies the lower frame coditio. The the implicatios (i) (ii) (iii) hold. I geeral, statemet (iii) does ot imply ay of the other statemets, but if T is closed ad surjective, the all statemets are equivalet. Proof. (i) (ii): By Lemma./(ii), the aalysis operator U is surjective, ad sice {f i } is complete, it is also ijective. From Lemma 3.1 it follows that {f i } satisfies the lower frame coditio. That {f i} is miimal follows easily from the defiitio of Riesz-Fischer sequeces. (ii) (iii): Suppose c if i = 0 with ot all c i zero. The there is some j such that c j 0 ad hece f j = c i i j c j f i, implyig f j spa {f i } i j, cotradictig the miimality of {f i }. We ow show that (iii) does ot imply (ii). I Theorem 3.5 below we will show that i a arbitrary Hilbert space there exists a ω-idepedet sequece {f i } which satisfies the lower frame coditio ad for which there is a f H such that o sequece of scalars {a i } satisfies f = a if i. The {f i } {f} satisfies the lower frame coditio ad is ω-liearly idepedet, but is ot miimal, sice {f i } is already complete. Clearly, this argumet also shows that statemet (i) ca ot be satisfied. O the other had, if T is closed ad surjective, it is proved i [1] that there exists a Bessel sequece {g i } such that f = f, g i f i for all f H. Assumig statemet (iii), it follows that f i, g j = δ i,j, i.e. {g i } is a biorthogoal Bessel sequece; thus, via Propositio.3, {f i } is a Riesz-Fischer sequece, ad completeess of it follows from the lower frame boud Riesz-Fischer sequeces ca also be characterized by the followig property, ivolvig lower frame bouds for the subspaces spaed by fiite subsets. Propositio 3.3. Let {f i } H, ad let {I } =1 be a family of fiite subsets of N such that I N. Deote by A opt I the optimal lower frame boud for {f i } i spa {f i }. The {f i } is a Riesz-Fischer sequece if ad oly if it is (fiitely) liearly idepedet ad if N A opt I > 0. The proof for this propositio follows the same lies as [3: Propositio 1.1] where the statemet was proved uder the additioal coditio that {f i }
Riesz-Fischer Sequeces 309 was a frame for H. Uder this extra coditio, the characterizatio was first proved by Kim ad Lim [4] as a cosequece of a series of Theorems. The propositio below characterizes sequeces satisfyig the lower frame coditio i terms of a expasio property. Propositio 3.4. Let {f i } H. The {f i} satisfies the lower frame coditio if ad oly if there exists a Bessel sequece {g i } H such that f = f, f i g i, f D(U). (3.1) Proof. Assume that {f i } satisfies the lower frame coditio. The U 1 : R(U) H is bouded. Defie a liear operator V : l (N) H by V = U 1 o R(U) ad V = 0 o R(U) ad extedig it liearly. The V is bouded. Let {e i } be the caoical basis for l (N) ad set g i = V e i. The {g i } is a Bessel sequece ad, by costructio, for all f D(U) we have f = V Uf = f, f i g i. O the other had, if {g i } is a Bessel sequece with boud B ad (3.1) is satisfied, the for all f D(U) f = f, f i g i B f, f i meaig that the lower frame coditio is satisfied Note that whe {f i } satisfies the lower frame coditio, the Bessel sequece {g i } costructed i the proof of Propositio 3.4 belogs to D(U). Observe that equality (3.1) might hold for all f H without D(U) beig equal to H. For istace, if {e i } is a orthoormal basis ad we defie f i = ie i (i N), the D(U) = { f = c i e i which is oly a subspace of H. Nevertheless, } ic i < f = f, f i 1 i e i, f H. (3.) Note that {ie i } is a Riesz-Fischer sequece, but ot a Riesz basis. For several families of elemets havig a special structure, the Riesz-Fischer
310 P. Casazza et al. property implies the upper Riesz basis coditio; let us just metio families of complex expoetials i L ( π, π) (cf. [5, 7, 8]). As far as we kow, o example of a orm-bouded family i a geeral Hilbert space satisfyig the Riesz-Fischer property but ot the upper Riesz basis coditio has bee kow. Theorem 3.5 will provide such a example. As we have see i Propositio 3.4, the lower frame coditio o {f i } is eough to obtai a Bessel sequece {g i } such that (3.1) holds. I (3.) we have see that represetatio (3.1) might hold for all f H, eve if D(U) is a proper subspace of H; oe could hope that the represetatio always hold o H. Our ext purpose is to prove that this is ot the case. We eed to do some preparatio before the proof, but we state the result already ow. Theorem 3.5. I every separable, ifiite dimesioal Hilbert space H there exists a orm-bouded Riesz-Fischer sequece {f i } for which the followig statemets are true: (1) {f i } has lower frame boud 1 ad o fiite upper frame boud. () D(U) is dese i H, ad {f i } D(U). (3) {f i } is ω-idepedet. (4) {f i } is ot a (Schauder) basis for H. (5) There is a f H so that, for o sequece of scalars {a i }, f = i a if i. (6) There is o family of fuctios {g i } so that, for every f H, f = i f, f i g i. Moreover, statemets (4) - (6) hold for all permutatios of {f i }. Our proof of Theorem 3.5 is costructive, ad the result was used i the proof of Theorem 3. to show that i geeral statemet (iii) does ot imply statemet (i). The idea i the costructio provig Theorem 3.5 is to cosider a Hilbert space H which is a direct sum of subspaces of icreasig order. Before we go ito details with the costructio, we eed some prelimiary results. Give N, let H be a Hilbert space of dimesio ad let {e i } be a orthoormal basis of H. Let P be the orthogoal projectio oto the uit 1 vector e i, i.e. ( ) P a i e i = a i e i. Let H 1 = (I P )H. For all 1 j 1 let fj = e j e. Note that {fj } 1 j=1 is a liearly idepedet family which spas H1. Our first lemma will idetify the frame bouds ad the dual frame for subfamilies of {fj } 1 j=1.
Riesz-Fischer Sequeces 311 Lemma 3.6. Give ay N ad ay I {1,..., 1}, the family {fj } is a liearly idepedet frame for its spa with lower frame boud 1 (which is optimal for I > 1) ad upper frame boud at least I +3. The dual frame for {fj } 1 j=1 is give by g j = 1 e j 1 e i, j = 1,..., 1. Proof. Give f spa {f j }, there are scalars a j so that i j f = a j f j = ( a j e j a i )e. (3.3) Note that f = a j + a i ad f, f j = a j + a i. Thus f, fj = a j + a i = [ a j + ][ a i a j + ] a i Here we observe that Thus = a j + ( [ Re a j ]) a i + I a i. ( [ ]) Re a j a i = Re [ ] [( )[ ]] a j a i = Re a j a i = f, fj = a j + ( I + ) a i = f + ( I + 1) a i. a i. So the choice A = 1 is a lower frame boud. If I > 1, we ca choose {a i } such that a i = 0, so the choice f = a if i with exactly those
31 P. Casazza et al. coefficiets shows that A = 1 is actually the optimal lower boud i this case. If I = 1, say, I = {j}, the relatio (3.3) betwee f ad {a i } gives ( I + 1) a i = a j = f fj = f ad so the optimal lower boud is A = i this case. Now we fix i I ad compute e i e, fj = 4 + I 1 = I + 3 = I + 3 e i e. It follows that the optimal upper boud is at least I +3. Sice our family {fj } 1 j=1 is liearly idepedet, the dual frame {g j} 1 j=1 is the family of dual fuctioals for the (Schauder) basis {fj } 1 j=1. We will ow compute this family explicitly. Because of symmetry, it suffices to fid g1 which we ow do. Write g1 = a ie i ad observe that g1 is uiquely determied by the followig 3 coditios: (i) 1 = g1, e 1 e = a 1 a. (ii) For all i 1, 0 = g1, fi = a i a. (iii) Sice g1 is i the orthogoal complemet of the vector e i, the coefficiets satisfy a i = 0. Now, by coditios (i) ad (ii) we have g1 = (1 + a )e 1 + a i= e i ad by coditio (iii) 1 + a + ( 1)a = 0. Hece, 1 = a, ad so a = 1. Fially, a 1 = 1 + a = 1 Recall that the basis costat K for a sequece {f i } i H is defied as { } m K = sup c if i c if i : 1 m < ; c 1,..., c C, c i f i 0 (for fiite sequeces {f i } N we replace < by N ). To make the calculatios i the ext lemma easier, we will work with H+1. 1 Lemma 3.7. Let N ad σ be a permutatio of {1,,..., }. The there is a sequece of scalars {a i } so that a i f +1 = + 1 while a i f +1 I particular, the basis costat for {f +1 } is at least +1. =.
Riesz-Fischer Sequeces 313 Proof. Let a i = { 1 for 1 i 1 for + 1 i. The a i f +1 Also, a i = 0 implies = a i + a i = 1 +. Hece, ad the statemet is proved a i f +1 = a i f +1 = a i e. a i = It is proved i [6] that {f i } ca oly be a basis if the basis costat is fiite. We are ow ready for the proof of Theorem 3.5. Proof of Theorem 3.5. Usig the otatio above we cosider the Hilbert space ( ) H = = H 1 We refer to [6] for details about such costructios. Let the sequece {f i } be ay eumeratio of {fj } 1, j=1,=. Sice {f j } 1 j=1 spas H1 ad is liearly idepedet for each =, 3,..., statemets (1) ad (3) follow. Statemet () is clear. We ow prove that {f i } ca ot be a Schauder basis; sice {f i} is defied as a arbitrary eumeratio of the elemets i {fj } 1, j=1,=, this will prove statemet (4). The basis costat for {f i } is larger tha or equal to the basis costat for ay subsequece. But for each N, a permutatio of the family {f +1 j } j=1 is a subsequece of {f i}, ad by Lemma 3.7 its basis costat is at least ( + 1)/; thus the basis costat for {f i } is ifiite, ad it ca ot be a basis. This proves statemet (4). We ow prove statemet (5). It clearly follows from statemet (3) that wheever f H, if there is a sequece of scalars {a j } so that f = j a jf j, the {a j } is uique. Sice {f j } is ot a Schauder basis, this gives statemet (5). l.
314 P. Casazza et al. For the proof of statemet (6) we observe that correspodig to {fj } 1, j=1,= the dual fuctioals {gj } 1, j=1,= are by Lemma 3.6 give by g j = 1 e j 1 e i for 1 j 1, =, 3,.... i j This family is the oly cadidate to satisfy statemet (6). I fact, suppose that a sequece {h j } 1, j=1,=1 satisfies f = j, f, f j h j for all f H. Now, for all m ad all 1 i 1, gj m, f i = 0. Also, gm j, f i m = 0 for all 1 i j m 1 while gj m, f j m = 1. Puttig this altogether, g m j = i, g m j, f i h i = g m j, f m j h m j = h m j. That is, h m j = gj m for all m N ad all 1 j m 1. Now we observe that this family does ot work for recostructio. For N, {gj } 1 j=1 are the dual fuctioals to {fj } 1 j=1. Sice {f j } 1, j=1,=1 is ot a basis, we coclude that {gj } 1, j=1,=1 is ot a basis. Sice {g j } 1, j=1,=1 is clearly a ω-idepedet family, this meas that there exists f H which ca ot be writte f = j, c j g j for ay choice of coefficiets {c j }. This proves statemet (6) To coclude the paper we observe that if every subfamily of {f i } satisfies the lower frame coditio with a commo boud A, the there exists a subfamily of {f i } which satisfies the lower Riesz basis coditio. The proof is similar to that of [: Theorem 3.]. Propositio 3.8. Suppose that {f i } satisfies (1.1) ad that every subfamily {f i } (J I) satisfies A f i J f, f i f spa {f i }. (3.9) The {f i } cotais a complete subfamily {f i } for which A c i c i f i i J i J (3.10) for all fiite sequeces {c i } i J. I Propositio 3.8 the coclusio A i J c i i J c if i actually holds for all sequeces {c i } l for which c i f i is coverget.
Riesz-Fischer Sequeces 315 Refereces [1] Christese, O.: Frames ad pseudo-iverses. J. Math. Aal. Appl. 195 (1995), 401 414. [] Christese, O.: Frames cotaiig a Riesz basis ad approximatio of the frame coefficiets usig fiite dimesioal methods. J. Math. Aal. Appl. 199 (1996), 56 70. [3] Christese, O. ad A. M. Lider: Frames of expoetials: lower frame bouds for fiite subfamilies ad approximatio of the iverse frame operator. Li. Alg. Appl. 33 (001), 117 130. [4] Kim, H. O. ad J. K. Lim: New characterizatios of Riesz bases. Appl. Comp. Harm. Aal. 4 (1997), 9. [5] Lider, A.: A uiversal costat for expoetial Riesz sequeces. Z. Aal. Aw. 19 (000), 553 559. [6] Lidestrauss, J. ad L. Tzafriri: Classical Baach Spaces. Part I: Sequece Spaces. New York - Heidelberg: Spriger-Verlag 1977. [7] Youg, R. M.: A Itroductio to Noharmoic Fourier Series, d. ed. New York: Academic Press 001. [8] Youg, R. M.: Iterpolatio i a classical Hilbert space of etire fuctios. Tras. Amer. Math. Soc. 19 (1974), 97 114. Received 6.09.001