BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES PETER G. CASAZZA AND M. C. LAMMERS Astrct. We provide detiled development of the L 1 function vlued inner product on L 2 (R) known s the rcket product. In ddition to some of the more sic properties, we show tht this inner product hs Bessel s inequlity, Riesz Representtion Theorem, nd Grm-Schmidt process. We then pply this to Weyl-Heisenerg frmes to show tht there exist compressed versions of the frme opertor, the frme trnsform nd the prefrme opertor. Finlly, we introduce the notion of n -frme nd show tht there is n equivlence etween the frmes of trnsltes for this function vlued inner product nd Weyl-Heisenerg frmes. 1. Introduction While working on some deep questions in non-hrmonic Fourier series, Duffin nd Scheffer [7] introduced the notion of frme for Hilert spces. Outside of this re, this ide seems to hve een lost until Duechies, Grossmn nd Meyer [5] rought ttention to it in 1986. They show tht Duffin nd Scheffer s definition ws n strction of concept introduced y Gor [8] in 1946 for doing signl nlysis. Tody the frmes introduced y Gor re clled Gor frmes or Weyl-Heisenerg frmes nd ply n importnt role in signl nlysis. In the study of shift invrint systems nd frmes severl uthors, including de Boor, DeVore, Ron nd Shen [1, 2, 13, 14], hve mde extensive use of the so clled rcket product [f, g](x) f(x + β)g(x + β). β 2πZ d This turns out to e specil cse of n inner product for Hilert C module used effectively y Rieffel nd others to produce results in the field of hrmonic nlysis on non-commuttive groups. For reference on this, we refer the reder to [12]. In wht follows we give more thorough development of 1991 Mthemtics Suject Clssifiction. Primry: 42A65, 42C15, 42C3. Key words nd phrses. Weyl-Heisenerg (Gor) frmes, inner products, Hilert C modules. The first uthor ws supported y NSF DMS 97618. 1

2 PETER G. CASAZZA AND M. C. LAMMERS the rcket product on L 2 (R) nd its ppliction to univrite principl Weyl- Heisenerg systems. Becuse we would like to chnge the shift prmeter to ritrry R + we will refer to this rcket product s the -inner product. Let us outline the orgniztion of the pper. In section 2 we review some of the fundmentls regrding Weyl-Heisenerg systems. In section 3 we give good reference for the rcket product nd some of its sic properties. In section 4 we develop the orthogonlity of this function vlued inner product including the notions of Bessel s inequlity nd Grm-Schmidt process. In section 5 we exmine the opertors ssocited with this inner product nd prove two Riesz Representtion Theorems. Finlly, in section 6 we pply these notions to Weyl-Heisenerg systems. Here we show tht ll the opertors ssocited to Weyl-Heisenerg system hve compression with regrd to this function vlued inner product nd relte this to the Ron nd Shen theory. We go on to introduce the notion of frme for this function vlued inner product nd show tht frmes of trnsltes here coincide with Weyl-Heisenerg frmes. the uthors would like to thnk the referee s nd A.J.E.M. Jnssen whose recommendtions gretly improved this mnuscript. 2. Preliminries We use N, Z, R, C to denote the nturl numers, integers, rel numers nd complex numers, respectively. A sclr is n element of R or C. Integrtion is lwys with respect to Leesgue mesure. L 2 (R) will denote the complex Hilert spce of squre integrle functions mpping R into C. A ounded unconditionl sis for Hilert spce H is clled Riesz sis. Tht is, (f n ) is Riesz sis for H if nd only if there is n orthonorml sis (e n ) for H nd n invertile opertor T : H H defined y T (e n ) f n, for ll n. We cll (f n ) Riesz sic sequence if it is Riesz sis for its closed liner spn. For E H, we write spn E for the closed liner spn of E. In 1952, Duffin nd Scheffer [7] introduced the notion of frme for Hilert spce. Definition 2.1. A sequence (f n ) of elements of Hilert spce H is clled frme if there re constnts A, B > such tht (2.1) A f 2 f, f n 2 B f 2, for ll f H. The numers A, B re clled the lower nd upper frme ounds respectively. The lrgest numer A > nd smllest numer B > stisfying the frme inequlities for ll f H re clled the optiml frme ounds. The frme is tight frme if A B nd normlized tight frme if

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 3 A B 1. If f n H, for ll n Z, we cll (f n ) frme sequence if it is frme for its closed liner spn in H. We will consider frmes from the opertor theoretic point of view. To formulte this pproch, let (e n ) e n orthonorml sis for n infinite dimensionl Hilert spce H nd let f n H, for ll n Z. We cll the opertor T : H H given y T e n f n the prefrme opertor ssocited with (f n ). Now, for ech f H nd n Z we hve T f, e n f, T e n f, f n. Thus T f n f, f n e n, nd T f 2 n f, f n 2, for ll f H. It follows tht the prefrme opertor is ounded if nd only if (f n ) hs finite upper frme ound B. Compring this to Definition 2.1 we hve Theorem 2.2. Let H e Hilert spce with n orthonorml sis (e n ). Also let (f n ) e sequence of elements of H nd let T e n f n e the prefrme opertor. The following re equivlent: (1) (f n ) is frme for H. (2) The opertor T is ounded, liner nd onto. (3) The opertor T is n (possily into) isomorphism clled the frme trnsform. Moreover, (f n ) is normlized tight frme if nd only if the prefrme opertor is quotient mp (i.e. co-isometry). It follows tht S T T is n invertile opertor on H, clled the frme opertor. Moreover, we hve Sf T T f T ( n f, f n e n ) n f, f n T e n n f, f n f n. A direct clcultion now yields Sf, f n f, f n 2. Therefore, the frme opertor is positive, self-djoint invertile opertor on H. Also, the frme inequlities (2.1) yield tht (f n ) is frme with frme ounds A, B > if nd only if A I S B I. Hence, (f n ) is normlized tight frme if nd only if S I. We will work here with prticulr clss of frmes clled Weyl-Heisenerg frmes. To formulte these frmes, we first need some nottion. For function f on R we define the opertors:

4 PETER G. CASAZZA AND M. C. LAMMERS Trnsltion: T f(x) f(x ), R Modultion: E f(x) e 2πix f(x), R Diltion: D f(x) 1/2 f(x/), R {} We lso use the symol E to denote the exponentil function E (x) e 2πix. Ech of the opertors T, E, D re unitry opertors on L 2 (R). In 1946 Gor [8] formulted fundmentl pproch to signl decomposition in terms of elementry signls. This method resulted in Gor frmes or s they re often clled tody Weyl-Heisenerg frmes. Definition 2.3. If, R nd g L 2 (R) we cll (E m T n g) m, Weyl- Heisenerg system (WH-system for short) nd denote it y (g,, ). We denote y (g, ) the fmily (T n g). We cll g the window function. If the WH-system (g,, ) forms frme for L 2 (R), we cll this Weyl- Heisenerg frme (WH-frme for short). The numers, re the frme prmeters with eing the shift prmeter nd eing the modultion prmeter. We will e interested in when there re finite upper frme ounds for WH-system. Tht is, we wish to know when the system (g,, ) is Bessel system lso referred to s prefrme functions. We denote this clss y PF. It is esily checked tht Proposition 2.4. The following re equivlent: (1) g PF. (2) The opertor Sf n,m f, E m T n g E m T n g, is well defined ounded liner opertor on L 2 (R). We will need the WH-frme identity due to Duechies [4]. To simplify the nottion little we introduce the following uxiliry functions. Define for g L 2 (R) nd ll k Z G k (t) g(t n)g(t n k/). In prticulr, G (t) g(t n) 2. Theorem 2.5. (WH-frme Identity.) If n g(t n) 2 f L 2 (R) is ounded nd compctly supported, then f, E m T n g 2 F 1 (f) + F 2 (f), n,m Z B.e. nd

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 5 where nd F 1 (f) 1 f(t) 2 G (t) dt, R F 2 (f) 1 f(t)f(t k k R )G k(t) dt 1 2Re f(t)f(t k )G k(t) dt. k 1 R There re mny restrictions on the g,, in order tht (g,, ) form WHfrme. We will mke note of few of them here. The first is simple ppliction of the WH-frme Identity. Tht is, if we put functions supported on [, 1 ] into this identity, then F 2 (f). Now the WH-frme Identity comined with the frme condition quickly yields, Theorem 2.6. If (g,, ) is WH-frme with frme ounds A, B then A 1 G (t) B,.e. A sic result of Ron nd Shen [14] yields similr upper ound condition with replced y 1. Proposition 2.7. (g,, ) is WH-frme with frme ounds A,B, iff the opertor K(x) : {g(x n j/)} n,j stisfies AI KK (x) BI. Hence if (g,, ) hs finite upper frme ound, then g(t n ) 2 B.e. A recent very importnt result ws proved independently y Duechies, H. Lndu nd Z. Lndu [6], Jnssen [1], nd Ron nd Shen [14]. Theorem 2.8. For g L 2 (R) nd, R, the following re equivlent: (1) (g,, ) is WH-frme. (2) The fmily (E m T n g) m, is Riesz sic sequence in L 2 (R). We end these preliminries with rief discussion of the Ron nd Shen theory of Grmin nlysis for shift invrint systems from [13, 14]. At the hert of their technique is the pre-grmin opertor J nd its djoint J. Ron nd Shen show tht J is Fourier trnsform nlogue of the opertor T. In the Weyl-Heisenerg cse with 1 it tkes the form J (f) : ([f, T m ĝ]) m Z, where we consider T : L 2 (R) l 2 (Z). Then they clculte the dul Grmin G JJ which will correspond to Fourier trnsform version of wht we lter

6 PETER G. CASAZZA AND M. C. LAMMERS refer to s the compression of the frme opertor. In order to void confusion we point out sutle difference in our pproch. We use the prefrme nd frme opertors which re ounded opertors from L 2 (R) to L 2 (R) insted of the opertor T ove nd its dul. 3. Pointwise Inner Products A numer of the results elow cn e found in the erly ppers [1, 2, 13, 14]. For the ske of completeness, nd to crete good reference for this inner product for use in WH-systems we list them here. To gurntee tht our inner product is well defined, we need to first check some convergence properties for elements of L 2 (R). Proposition 3.1. For f, g L 2 (R) nd R + the series f(t n)g(t n) converges unconditionlly.e. to function in L 1 [, ]. Proof. If f, g L 2 (R) then fg L 1 (R). Hence, fg L 1 f(t n)g(t n) dt <. The Monotone Convergence Theorem yields oth the interchnge of the integrl nd the sum nd the existence of f(t n)g(t n) s function in L 1 [, ]. A simple ppliction of the Leesgue Dominted Convergence Theorem comined with Proposition 3.1 yields Corollry 3.2. For ll f, g L 2 (R) we hve f, g f(t n)g(t n) dt. Now we introduce the pointwise inner product for WH-frmes. We cn view this s function vlued inner product. Definition 3.3. Fix R +. For ll f, g L 2 (R) we define the -pointwise inner product of f nd g (clled the -inner product for short) y f, g (t) f(t n)g(t n), for ll t R. We define the -norm of f y f (t) f, f (t).

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 7 We emphsize here tht the -inner product nd the -norm re functions on R which re clerly -periodic. To cut down on nottion, whenever we hve n -periodic function on R, we will lso consider it function on [, ]. The convergence of these series is gurnteed y our erlier discussion. In fct, the -inner product <, > is n onto ( ut not 1-1) mpping from L 2 (R) L 2 (R) to the -periodic functions on R whose restriction to [, ] lie in L 1 [, ]. First we show tht the -inner product relly is good generliztion of the stndrd notion of inner products for Hilert spce. Since ll of these proofs follow directly from the definitions we omit them. Theorem 3.4. Let f, g, h L 2 (R), c, d C, nd, R. The following properties hold: (1) f, g is periodic function of period on R with f, g L 1 [, ]. (2) f L 2 (R) f (t) L 2 [,] (3) f, g f, g (t) dt. (4) cf + dg, h c f, h + d g, h. (5) f, cg + dh c f, g + d f, h. (6) f, g g, f. (7) fg, h f, gh, for fg, gh L 2 (R). (8) If f, g then f, g. (9) T f, T g T f, g. (1) T g 2 T g 2. (11) T f, g T f, T g. (12) f, g 1 D D 1 f, D 1 g 1. It is not difficult to mimic the stndrd proofs for the usul inner product on Hilert spce to otin the following results for the -inner product. Proposition 3.5. For ll f, g L 2 (R) the following hold.e.: (1) f, g f g. (2) f + g 2 f 2 + 2Re f, g + g 2. (3) f + g f + g. (4) f + g 2 + f g 2 2( f 2 + g 2 ). Since our -inner product is n -periodic function, it enjoys some specil properties relted to -periodic functions.

8 PETER G. CASAZZA AND M. C. LAMMERS Proposition 3.6. Let f, g L 2 (R) nd let h L (R) e n -periodic function. Then fh, g h f, g nd f, hg h f, g. In prticulr, if h stisfies h(t).e., then f, g fh, g f, gh. Proof. We compute fh, g (t) f(t n)h(t n)g(t n) f(t n)h(t)g(t n) if nd only if h(t) f(t n)g(t n) h(t) f, g (t). Next we normlize our functions in the -inner product. For f L 2 (R), we define the -pointwise normliztion of f to e { f(t) N (f)(t) : f f (t) (t) : f (t). We now hve Proposition 3.7. Let f, g L 2 (R). (1) We hve N (f), g f, g f (t), where f (t). In prticulr, f, g (t) if nd only if N (f), g (t). (2) For f.e. we hve (3) We hve N (f), N (f) (t) where λ denotes Leesgue mesure. (4) N (N (f)) N (f). Proof. (1) We compute N (f), g N (f)(t n) 2 1,.e. N (f) 2 L 2 (R) λ(supp f [,] (t)). N (f)(t n)g(t n) f(t n) g(t n). f (t n)

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 9 Since our inner product is -periodic, this equlity ecomes, 1 f(t n)g(t n) f, g (t) f (t) f (t), where f (t). (2)-(4) re strightforwrd clcultions. 4. -orthogonlity The notion of orthogonlity with respect to the -inner product hs een used primrily to descrie the orthogonl complement in the usul inner product for shift-invrint spces. In this section we explore more thoroughly wht it mens to e -orthogonl nd develop such things s -orthonorml sequences nd Bessel inequlity for the -inner product. This property gives one of the min pplictions of the -inner product in Weyl-Heisenerg frme theory. For s we will see, orthogonlity in this form is very strong. Definition 4.1. For f, g L 2 (R), we sy tht f nd g re -orthogonl, nd write f g, if f, g (t).e.. We define the -orthogonl complement of E L 2 (R) y E {g : f, g, for ll f E}. Similrly, n -orthogonl sequence is sequence (f n ) stisfying f n f m, for ll n m. This is n -orthonorml sequence if we lso hve f n 1.e. We now identify n importnt clss of functions for working with the -inner product. Definition 4.2. We sy tht g L 2 (R) is -ounded if there is B > so tht g, g (t) B, for.. t We let L (R) denote the fmily of -ounded functions. We hve tht L (R) is non-closed (in the L 2 (R) norm) liner suspce of L (R). Note lso tht the Wiener mlgm spce is suspce of L (R). We hve not defined orthonorml ses for the -inner product yet since, s we will see, this requires little more cre. First we need to develop the sic properties of -orthogonlity. Proposition 4.3. If E L 2 (R) nd B {f : f L (R) nd f is -periodic} then E φ B (φe) (spn φ B φe).

1 PETER G. CASAZZA AND M. C. LAMMERS Proof. Let f E. For ny g E nd ny -periodic function φ B we hve y Proposition 3.6 f, φg (t) φ(t) f, g (t). Hence, f φg. Tht is, f (φe). Now let f (φe), the intersection eing tken over ll ounded -periodic φ. Let g E nd define for n N, { f, g (t) : f, g φ n (t) (t) n : otherwise. Note tht φ n is -periodic. Now we compute, f, φ n g f(t)φ n (t)g(t)dt R ( ) f(t n)g(t n) φ n (t) dt f, g (t)φ n (t) dt φ n (t) 2 dt. Therefore, φ n, for ll n Z. Hence, f, g (t).e., nd so f g. Tht is, f E. By Theorem 3.4 (8), we hve tht E E. Corollry 4.4. For E L 2 (R), E is norm closed liner suspce of E. The next result which cn e found in [1] shows more clerly wht orthogonlity mens in this setting. Proposition 4.5. For f, g L 2 (R), the following re equivlent: (1) f g. (2) spn m Z E m f spn m m ZE g. Proof. Fix m Z nd compute f, E m g f, g (t)e 2πi( m )t dt < f, g > (m). It follows tht f, E m g, for ll m Z if nd only if ll the Fourier coefficients of < f, g > (t) re zero. A moment s reflection should convince the reder tht this is ll we need. Definition 4.6. We sy tht E L 2 (R) is n -periodic closed set if for ny f E nd ny φ L (R) we hve tht φf E. The next result follows immeditely from Propositions 4.3 nd 4.5.

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 11 Corollry 4.7. For ny E L 2 (R), E n -periodic closed set then E E. is n -periodic closed set. If E is Now we oserve wht orthogonlity mens for (E m g) in terms of the regulr inner product. Proposition 4.8. If g L 2 (R) nd g 1.e., then ( 1 E m g) m Z is n orthonorml sequence in L 2 (R). Proof. For ny n, m Z we hve E n g, E m g g(t) 2 e 2πi[(n m)/]t dt R g 2 (t)e 2πi[(n m)/]t dt e 2πi[(n m)/]t dt δ nm. Corollry 4.9. If (g n ) n N is n -orthonorml sequence in L 2 (R), then (E m g n) n,m Z is n orthonorml sequence in L 2 (R). Proof. We need tht for ll (n, m) (l, k) Z Z, E m g n E k/ g l. But, if n l, this is Proposition 4.5, nd if n l, this is Proposition 4.8. Corollry 4.9 tells us how to define n -orthonorml sis. Definition 4.1. Let g n L 2 (R). We cll (g n ) n -orthonorml sis for L 2 (R) if it is n -orthonorml sequence nd spn (E m g n) n,m Z L 2 (R). Proposition 4.11. A sequence (g n ) in L 2 (R) is n -orthonorml sis if nd only if (E m g n) n,m Z is n orthonorml sis for L 2 (R). We would like to cpture the importnt Bessel s Inequlity for -orthonorml sequences ut efore we do so we need to insure tht f, g g remins in L 2 (R) for functions g L (R). Proposition 4.12. If g, h L (R) then f, g h L 2 (R) for ll f L 2 (R). Proof. First we need to show f, g L 2 ([, ]). Let B esssup [,) g 2 (t) nd C esssup [,) h 2 (t). This follows from the Cuchy-Schwrz inequlity for the -inner product:

12 PETER G. CASAZZA AND M. C. LAMMERS f, g (t) 2 L 2 [,] B f, g (t) 2 dt f, f (t) g, g (t) dt f, f (t) dt B f 2 L 2 (R). Now we cn prove the proposition using the Monotone Convergence Theorem nd the result ove: f, g h 2 L 2 (R) R n f, g (t)h(t) 2 dt f, g (t) 2 h(t n) 2 dt f, g (t) 2 h, h (t)dt BC f 2 L 2 (R) Theorem 4.13. Let (g n ) n N e n -orthonorml sequence in L 2 (R) nd f L 2 (R). (1) the series of functions n N f, g n g n converges in L 2 (R). (2) We hve Bessel s Inequlity, f, f f, g n 2. n1 Note tht this is n inequlity for functions. Moreover, if f spn (E m g n) m,, then f, f f, g n 2. n1 Proof. First we note tht the g n re in L (R) so ech f, g n g n is in L 2 (R). Fix 1 m nd let m h f, g n g n. n1

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 13 Using the fct tht the -inner product of two functions is -periodic (nd hence my e fctored out of the -inner product) we hve m m h, h f, g n g n, f, g k g k n1 m k1 f, g n f, g k g n, g k n,k1 n1 m f, g n 2. Letting g f h we hve y the sme type of clcultion s ove, m m h, g f, g n g n, f f, g k g k n1 m f, g n 2 n1 k1 k1 m f, g k 2. So we hve decomposed f into two -orthogonl functions h, g. Therefore, f, f h + g, h + g h, h + g, g m f, g n 2 + g, g n1 m f, g n 2. Since m ws ritrry, we hve (2) of the Theorem. For (1), we just put together wht we know. By (2) nd the Monotone Convergence Theorem, we hve tht the series of functions n N f, g n 2 converges in L 1 [, ]. But, y our clcultions ove nd the properties of the -norm, m f, g n g n 2 L 2 (R) nk n1 m f, g n g n 2 (t) dt nk m f, g n g n, nk m f, g n g n nk m f, g n 2 (t) dt. nk (t) dt Now, n N f, g n 2 converges in L 1 [, ] implies tht the right hnd side of our equlity goes to zero s k.

14 PETER G. CASAZZA AND M. C. LAMMERS We end this section with Grm-Schmidt process for the -inner product. First we need result which shows tht this process produces functions which re in the proper spns. Proposition 4.14. Let f, g, h L 2 (R). We hve: (1) N (g) spn (E m g) m Z. (2) If ny two of f, g, h re in L (R), then f, g h spn (E m g) m Z. Proof. (1): For ech n N let Also, let E n {t [, ] : g, g (t) 2 n or g, g (t) 1 n }. Since g L 2 (R), we hve g 2 Ẽ n m Z (E n + m). g, g (t) dt <. Hence, lim n λ(e n ). Let F n [, ] E n nd F n m Z (F n + m). Now, Hence, Hence, 1 n 1 χ Fn g, χ Fn g 1 χ Fn g, χ Fn g n. L (R). χ Fn g χ Fn g, χ Fn g + χẽn g spn (E m g) m Z. Also, χ Fn g χ Fn g, χ Fn g + χẽn g N (g) L 2 (R) χẽn g χ Ẽ n g g, g χẽn g + χ Ẽ n g g, g ( 1/2 χẽn g dt) 2 + N (χẽn g) R ( ) 1/2 g, g (t) dt + λ(e n ). E n

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 15 But the right hnd side of the ove inequlity goes to zero s n. (2): Assume first tht f, g L (R). Let B esssup [,) f 2 (t) nd C esssup [,) g 2 (t). Now y Cuchy-Schwrz f, g (t) f (t) g (t) B C. Therefore, f, g is ounded -periodic function on R. This implies tht f, g h L 2 (R). Now suppose tht g, h L (R). Then f, g h L 2 (R) y Proposition 4.12. A direct clcultion shows spn (E m g) m Z {φg : φ is -periodic nd φg L 2 (R)}. So y the ove, we hve tht f, h g spn (E m g) m Z. Definition 4.15. Let g n L 2 (R), for 1 n k. We sy tht (g n ) k n1 is - linerly independent if for ech 1 n k, g n / spn (E m g i) m Z;1 i n k. An ritrry fmily is -linerly independent if every finite su-fmily is -linerly independent. Now we crry out the Grm-Schmidt process. Theorem 4.16. (Grm-Schmidt ortho-normliztion procedure) Let (g n ) n N e n -linerly independent sequence in L 2 (R) for >. Then there exists n -orthonorml sequence (e n ) n N stisfying for ll n N: spn (E m g k) m Z,1 k n spn (E m e k) m Z,1 k n. Proof. We proceed y induction. First let e 1 N (g 1 ). If (e i ) n i1 hve een defined to stisfy the theorem, let e n+1 N (g n+1 n g i, e i e i ) nd h g n+1 i1 n g n+1, e i e i. Note tht h y our -linerly independent ssumption nd Proposition 4.14. Now, for 1 k n we hve e n+1, e k ( ) 1 n g n+1, e k h, h g n+1, e i e i, e k i1 1 ( g n+1, e k h, h g n+1, e k e k, e k ). i1 The sttement out the liner spns follows from Proposition 4.14.

16 PETER G. CASAZZA AND M. C. LAMMERS 5. -Fctorle Opertors Now we consider opertors on L 2 (R) which ehve nturlly with respect to the -inner product. We will cll these opertors -fctorle opertors. Definition 5.1. Fix 1 p. We sy tht liner opertor L : L 2 (R) L p (E) is n -fctorle opertor if for ny fctoriztion f φg where f, g L 2 (R) nd φ is n -periodic function on R we hve L(f) L(φg) φl(g). First we show it is enough to consider fctoriztions over L ([, )) nd see these re exctly the opertors tht commute with ll E m Proposition 5.2. Let L e ounded opertor from L 2 (R) to L 2 ([, )). Then L is -fctorle if nd only if L(φf) φl(f) for ll f L 2 (R) nd ll - periodic φ L (R). Proof. Assume φ is -periodic, f, g L 2 (R) nd f φg. For ll n N let Let E n [, 1] F n nd Now, F n {t [, ] : φ(t) > n}. Ẽ n m Z (E n + m) nd χẽn φg φg 2 L 2 (R) R F n m Z (F n + m). χ Fn φ(t)g(t) 2 dt χ Fn φ(t) 2 g, g (t) dt. Since φg L 2 (R) nd lim n λ(f n ), it follows tht h n : χẽn φg converges to φg in L 2 (R). Since L is ounded liner opertor, it follows tht L(h n ) converges to L(φg). But, L(h n ) χẽn φl(g) y our ssumption. Now, L(h n ) L h n L φg L f. Finlly, since L(h n ) φl(g) it follows from the Leesgue Dominted Convergence Theorem tht φl(g) L 2 (R) nd L(h n ) φl(g). This completes the proof of the Proposition. We hve immeditely Corollry 5.3. An opertor L : L 2 (R) L p (E) is -fctorle if nd only if L(E m g) E m L(g), for ll m Z. Tht is, L is -fctorle if nd only if it commutes with E m. Next we derive our first Riesz Representtion Theorem for -fctorle opertors. To simplify this proof s well s lter rguments we first prove lemm.

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 17 Lemm 5.4. Let L 1 nd L 2 e -fctorle opertors from L 2 (R) L 1 [, ]. Then L 1 L 2 iff L 1 (f)(t)dt L 2 (f)(t)dt. Proof. Fix f L 2 (R). By our ssumption, for ll m Z we hve L 1 (E m L 1 (f)(t)e m (t)dt L 2 (f)(t)e m (t)dt L 2 (E m f)(t)dt. Hence, the Fourier coefficients for L 1 (f) nd L 2 (f) re the sme for ll f L 2 (R) nd therefore L 1 L 2 Our originl proof of the Riesz Representtion Theorem for -fctorle opertors ws cumersome. The direct proof elow ws kindly communicted to us y A.J.E.M. Jnssen. Theorem 5.5. ( Riesz Representtion Theorem 1): L is ounded -fctorle opertor from L 2 (R) to L 1 [, ] iff there is g L 2 (R) so tht L(f) f, g (t) for ll f L 2 (R). Moreover L g. Proof. Fix g L 2 (R) nd define L on L 2 (R) y L(f) f, g (t). Now for ny f L 2 (R) Lf f, g (t) L 1 [,] f(t n)g(t n) dt f(t n) 2 g(t n) 2 dt ( ) 1/2 ( ) 1/2 f(t n) 2 g(t n) 2 f L 2 (R) g L 2 (R). Letting g f we see tht L(g) g which, comined with the ove, shows tht L g. Assume L is n -fctorle opertor from L 2 (R) L 1 [, ]. Define the liner functionl Ψ on L 2 (R) y Ψ(f) L(f)(t)dt.

18 PETER G. CASAZZA AND M. C. LAMMERS By the stndrd Riesz Representtion Theorem, there is function g L 2 (R) so tht Ψ(f) f, g for ll f L 2 (R). Define the opertor L g (f) f, g (t). It follows tht Ψ(f) f, g f, g (t)dt L g (f)(t)dt L(f)(t)dt. Since L g nd L re -fctorle mps from L 2 (R) to L 1 [, ], they re equl y Lemm 5.4. Now, let L e ny -fctorle liner opertor from L 2 (R) to L p ([, ]), nd let E ker L. If f E, nd φ L (R), then L(φf) φl(f). So φf E. We summrize this elow. Proposition 5.6. If L is ny -fctorle liner opertor with kernel E, then E is n -periodic closed set nd so E E. One more property of -fctorle opertors into L 2 [, ] is tht the opertor is ounded pointwise with respect to the -norm. Proposition 5.7. Let L e n -fctorle liner opertor from L 2 (R) to L 2 [, ]. Then L is ounded if nd only if there is constnt B > (B L ) so tht for ll f L 2 (R) we hve L(f)(t) B f (t), for.e. t [, ]. Moreover, L is n isomorphism if nd only if there re constnts A, B > (A L 1 1, B L ) so tht for ll f L 2 (R) we hve A f (t) L(f)(t) B f (t), for.e. t [, ]. Proof. For ny ounded -periodic function φ on R, nd for every f L 2 (R) we hve φ(t) 2 L(f)(t) 2 dt It follows immeditely tht L(φf)(t) 2 dt L(φf) 2 L 2 ([,] L 2 φf 2 L 2 (R) L 2 L 2 φ(t) 2 f 2 (t) dt. R φ(t) 2 f(t) 2 dt L(f)(t) 2 L 2 f 2 (t), for.e. t [, ]. The other impliction is similr, s is the moreover prt of the Proposition. This gives us nother Riesz Representtion Theorem for opertors from L 2 (R) to L 2 [, ].

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 19 Theorem 5.8. (Riesz Representtion 2) L is ounded -fctorle opertor from L 2 (R) to L 2 [, ] iff there is g L (R) so tht L(f) f, g for ll f L 2 (R). Moreover L 2 ess sup [,] g, g. Proof. Let g e in L (R) nd define L to e L(f) f, g. The rest follows directly from the first prt of the proof of Proposition 4.12 nd gin, letting g f ove gives the norm of the opertor. Let L e ounded -fctorle opertor from L 2 (R) to L 2 [, ]. Since L 2 [, ] L 1 [, ] it is cler from Theorem 5.5 tht there exists g L 2 (R) so tht L(f) f, g, (t). By Proposition 5.7 we get g, g (t) g 2 (t) L(g)(t) g (t) L. Hence g (t) L.e. nd g L (R) We hve corresponding -norm ound for -fctorle opertors into L 2 (R). Proposition 5.9. If L : L 2 (R) L 2 (R) is n -fctorle opertor then L is ounded if nd only if there is constnt B > (A L ) so tht for ll f L 2 (R) we hve Lf (t) B f (t), for.e. t [, ]. Moreover, L is n isomorphism if nd only if there re constnts A, B > (A L 1 1, B L ) so tht for ll f L 2 (R) we hve A f (t) L(f)(t) B f (t), for ll t R. Proof. For ny f L 2 (R) nd ny ounded -periodic function φ we compute L(φf) 2 L 2 (R) L(φf)(t) 2 dt φ(t) 2 (Lf)(t) 2 dt R R φ(t) 2 (Lf)(t n) 2 dt φ(t) 2 Lf 2 (t) dt L 2 φf 2 L 2 (R) L 2 φ(t) 2 f(t) 2 dt L 2 φ(t) 2 f 2 (t) dt. It follows tht Lf 2 (t) L 2 f 2 (t), The rest of the proposition follows similrly. Note tht Proposition 5.9 shows tht -fctorle opertors must mp - ounded functions to -ounded functions. We end this section y verifying tht for -fctorle opertors L, the opertor L ehves s it should reltive to the -inner product. R.e.

2 PETER G. CASAZZA AND M. C. LAMMERS Proposition 5.1. If L is n -fctorle opertor from L 2 (R) to L 2 (R), then for ll f, g L 2 (R) we hve L(f), g (t) f, L (g) (t). Proof. Consider the opertor L(f) L(f), g (t) nd L (f) f, L (g) (t). Both of these re -fctorle opertors from L 2 (R) L 1 [, ]. Also, We re done y Lemm 5.4. L(f)(t)dt L(f), g f, L (g) L (f)(t)dt. 6. Weyl-Heisenerg frmes nd the -inner product Now we pply our -inner product theory to Weyl-Heisenerg frmes. This will produce compression representtions of the vrious opertors ssocited with frmes. We cll these compressions ecuse they no longer hve the modultion prmeter explicitly represented. Tht is, we re compressing the modultion prmeter into the 1/-inner product. We will lso relte our results to the Ron-Shen Theory [13, 14]. An excellent ccessile ccount of this theory (nd much more) cn e found in Jnssen s rticle [11]. This tretment is done for generl shift-invrint systems using only sic fcts from Fourier nlysis nd Leesgue integrtion. This is then pplied to Weyl-Heisenerg systems including representtions of the frme opertor s well s representtions nd clssifictions of the dul systems. For ny WH-frme (g,, ), it is well known tht the frme opertor S commutes with E m, T n. Thus, Corollry 5.3 yields: Corollry 6.1. If (g,, ) is WH-frme, then the frme opertor S is -fctorle opertor. 1 We next show tht the WH-frme Identity for (g,, ) hs n interesting representtion in oth the nd the 1 inner products. The known WH-frme identity requires tht the function f e ounded nd of compct support. While this remins condition for the WH-frme Identity derived from the -inner product we re le to extend this result to ll f L 2 (R) when we use the 1 -inner product. For this reson we present the theorems seprtely. The proof of oth these theorems hve their roots in the Heil nd Wlnut proof of the WH-frme Identity (see [9], Theorem 4.1.5). We refer the reder to Proposition 3.1 nd Corollry 3.2 for questions concerning convergence of the series nd integrls elow.

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 21 Theorem 6.2. Let g L (R), nd, R +. For ll f L 2 (R) which re ounded nd compctly supported we hve m, f, E m T n g 2 1 k T k f, f g, T k Proof. We strt with the WH-frme Identity relizing tht periodic. Hence g dt. g, T k g is - (6.1) m, f, E m T n g 2 1 k 1 k 1 k f(t)f(t k R ) g(t n)g(t n k )dt n f(t j)f(t k g, j) T k g dt j T k f, f g, T k g dt For the rest of this section we concentrte on the 1 inner product nd its reltionship to WH-frmes. In forthcoming pper on the WH-frme identity we show tht one my relx the condition on g. Tht is, the originl WH-frme identity holds for ll g L 2 (R) when f is ounded nd compctly supported. Theorem 6.3. Let g L (R), nd, R +. For ll f L 2 (R) we let f n f, T n g 1 then nd so n,m Z m Z f, E m T n g 2 f, T n g 1 2 f L 2 [, n 2 1 ] L 2 [, 1 ], f, E m T n g 2 f, T n g 1 2 f L 2 [, 1 ] n 2 L 2 [, 1 ].

22 PETER G. CASAZZA AND M. C. LAMMERS Proof. Since g L (R) we know ech f n L 2 [, 1 ] nd now we just compute f, E m T n g 2 f(t)g(t n)e 2πimt dt 2 m Z m Z R 1 m Z 1 m Z m Z 1 f(t k )g(t n k )e 2πimt dt 2 k Z 1 f, T n g 1 (t)e 2πimt dt 2 f n (m) 2 dt f n 2 f, T L 2 [, ng 1 ] 1 2 L 2 [, 1 ]. Now we wnt to directly relte our -inner product to WH-frmes. We egin with the compression we referred to ove. Proposition 6.4. If g, h L 1 (R), then for ll f L 2 (R) we hve f, E m g E m h 1 f, g h, 1 m Z where the series converges unconditionlly in L 2 (R). Hence, f, g 1 g spn (E m g) m Z. Proof. By our second Riesz Representtion Theorem 5.8 we know tht f, g 1 L 2 [, 1 ]. Next, for ny m Z we hve f, E m g 1 f, g 1 (t)e 2πimt dt f, g 1 (m). Therefore, if we restrict ourselves to L 2 [, 1 ] we hve f, g (m)e 2πimt 1 f, g 1 m Z f, E m g E m m Z Now y 4.12 we hve 1 f, g h L 2 (R) 1. There re mny interesting consequences of this proposition. First we recpture the following result due to de Boor, DeVore, nd Ron [1] Corollry 6.5. For g L 2 (R) nd R, the orthogonl projection P of L 2 (R) onto spn (E m g) m Z is P f 1 g 2 1 f, g 1 g,

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 23 where if g 1 (t) then g(t) so we interpret g(t) g 2 1 (t). Proof. By Proposition 4.9, we hve tht ( E m g 1 ) m Z is n orthonorml sequence in L 2 (R). Hence, for ll f L 2 (R) we hve y Proposition 6.4 P f f, g Em g E m g 1 g 1 m Z g g g g f, E m E m f, 1 g 1 g 1 g 1 g 1 g 2 1 m Z Comining Theorem 4.13 nd Corollry 6.5 we hve: g 1 f, g 1 Proposition 6.6. If (g n ) is 1 -orthonorml sequence in L2 (R), then P (f) f, g n 1 g n, is the orthogonl projection of L 2 (R) onto spn (E m g n ) n,m Z This llows us to compress the opertors ssocited with WH-system (g,, ). In [13] nd [14], Ron nd Shen mke use of the Dul Grmin to nlyze WHsystem (g, 1, ). This will correspond to compression of the frme opertor of the system (ĝ,, 1). Here we produce similr results for the frme opertor, prefrme opertor nd frme trnsform in the spce domin. Proposition 6.7. If g 1 opertor hve the following compressions 1 T (f) n B.e. then the frme trnsform nd the prefrme f, e n 1 T n g nd T (f) 1 where e n T n 1 [, 1 ), the stndrd 1 -orthonorml sis. Proof. If g 1 T (f) m,n B, y Proposition 6.4 we hve f, E m T n g E m T n (1 [, 1 ]) 1 n n f, T n g 1 e n f, T n g 1 e n, where we used the fct tht ( E m T n (1 [, 1 ])) is n orthonorml sis for L 2 (R). Hence T must e 1 fctorle which in turn implies T is. Continuing we get T (f) n T (f), e n 1 e n n f, T (e n ) 1 e n g.

24 PETER G. CASAZZA AND M. C. LAMMERS So T (e n ) 1 T ng nd the rest follows. Theorem 6.8. If (g,, ) is PF with frme opertor S, then S hs the form S(f) 1 f, T n g 1 T n g 1 P n f T n g 2 1, where P n is the orthogonl projection of L 2 (R) onto spn (E m T n g) m Z nd the series converges unconditionlly in L 2 (R). Proof. If (g,, ) is WH-frme then y Proposition 2.7 we hve tht g, g 1 B.e. Now, y definition of the frme opertor S we hve S(f) m, f, E m T n g E m T n g ( ) f, E m T n g E m T n g m Z 1 f, T n g 1 T n g. An ppliction of Corollry 6.5 nd Theorem 3.4 (1) completes the proof. We summrize some of the known results out normlized tight WH-frmes in the the following Proposition. These results re due to vrious uthors. Direct proofs from the definitions s well s historicl development my e found in [3]. Proposition 6.9. Let (g,, ) e WH-frme. the following re equivlent: (1) (E m T n g) n,m Z is normlized tight Weyl-Heisenerg frme. (2) ( 1 T n g) is n orthonorml sequence in the -inner product. (3) We hve tht g T k g, for ll k nd g, g.e. Putting Corollry 6.6 nd Proposition 6.9 together we hve Corollry 6.1. If (g,, ) is normlized tight Weyl-Heisenerg frme, then P (f) 1 f, T n g T n g is the orthogonl projection of L 2 (R) onto spn (E m T n g) n,m Z. Given the compressed representtion of the frme opertor it is now nturl to exmine the notion of frme nd Riesz sis with respect to pointwise inner product.

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 25 Definition 6.11. We sy tht sequence f n L 2 (R) is -Riesz sic sequence if there is n -orthonorml sis (g n ) nd n -fctorle opertor L on L 2 (R) with L(g n ) f n so tht L is invertile on its rnge. If L is surjective, we cll (f n ) -Riesz sis for L 2 (R). Proposition 6.12. Let f n L 2 (R), for ll n Z. The following re equivlent: (1) (f n ) is n -Riesz sic sequence. (2) (E m f n) is Riesz sic sequence. Proof. (1) (2): By ssumption, there is n -orthonorml sis (g n ) nd n -fctorle opertor L with L(g n ) f n for ll n Z. By the definition of n -orthonorml sis we hve tht ( 1 E m g n) m, is n orthonorml sis for L 2 (R). Since L is n isomorphism, it follows tht (L( 1 E m g n)) n,m Z ( 1 E m L(g n)) n,m Z ( 1 E m f n) n,m Z is Riesz sic sequence. (2) (1): Let g χ [,) so tht ( 1 E m T ng) m, is n orthonorml sis for L 2 (R). Then L( 1 E m T ng) E m f n is n -fctorle liner opertor which is n isomorphism ecuse (E m f n) is Riesz sic sequence. Hence, (f n ) is n -Riesz sic sequence. Corollry 6.13. Let f n L 2 (R), for ll n Z. The following re equivlent: (1) (f n ) is n -Riesz sis. (2) (E m f n) is Riesz sis for L 2 (R). Since the inner product on Hilert spce is used to define frme, we cn get corresponding concept for the -inner product. Definition 6.14. If g n L 2 (R), for ll n Z, we cll (g n ) n -frme sequence if there exist constnts A, B > so tht for ll f spn (E m g n) m, we hve A f 2 (t) f, g n (t) 2 B f 2 (t). If the inequlity ove holds for ll f L 2 (R) then we cll (g n ) n -frme. Now we hve the corresponding result to Theorem 2.2. Theorem 6.15. Let g n L 2 (R), for ll n Z. The following re equivlent: (1) (g n ) is n -frme. (2) If (e n ) is n -orthonorml sis for L 2 (R), nd T : L 2 (R) L 2 (R) with T (e n ) g n is -fctorle, then T is ounded, liner surjective opertor on L 2 (R).

26 PETER G. CASAZZA AND M. C. LAMMERS Proof. If T (e n ) g n, then T (f), e n f, T (e n ) f, g n. Hence, y Theorem 4.13 we hve tht T (f) f, g n e n nd T (f) 2 f, g n 2. Hence, (g n ) is n -frme sequence if nd only if A f 2 (t) T (f) 2 (t) B f 2 (t), for ll f L 2 (R). But, y Proposition 5.9, this is equivlent to T eing n isomorphism, which itself is equivlent to T eing ounded, liner onto opertor. Finlly, we cn relte this ck to our regulr frme sequences. Proposition 6.16. Let g n L 2 (R), for ll n Z. The following re equivlent: (1) (g n ) is n -frme sequence. (2) (E m g n) m, is frme sequence. Proof. (1) (2): If (g n ) is n -frme sequence, then there is n -orthonorml sis (e n ) for L 2 (R) nd n -fctorle onto (closed rnge) opertor T (e n ) g n. Now, (E m e n) n,m Z is n orthonorml sis for L 2 (R) nd T (E m e n) E m T (e n) E m g n. Hence, (E m g n) m, is frme sequence. (2) (1): Reverse the steps in prt I ove. The following Corollry is immedite from Theorem 6.15 nd Proposition 6.16. Corollry 6.17. Let g L 2 (R) nd, R. The following re equivlent: (1) (g, ) is 1 -frme. (2) (g,, ) is Weyl-Heisenerg frme. We conclude this pper with the following remrk out some ongoing reserch of Michel Frnk nd the two uthors. One cn show tht the spce L (R) my e viewed s the Leesgue spce L (l 2 ). If f, g L (R) then f, g (t) L [, ] insted of L 1 [, ]. This dense suspce of L 2 (R) equipped with the -inner product cn e viewed s Hilert C -Module where the rnge spce of the inner product is now the C -Alger L [, ]. The corollry ove shows tht there is strong connection etween frmes of trnsltes for this Hilert C -Module nd WH-frmes. This opens the door for pssing results ck nd forth etween WH-systems nd certin Hilert C - modules.

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES 27 References [1] C. de Boor, R. DeVore nd A. Ron,Approximtion from shift invrint suspces of L 2 (R d ), Trns. Amer. Mth. Soc., (1994) 341:787-86. [2] C. de Boor, R. DeVore nd A. Ron,The Structure of shift invrint spces nd pplictions to pproximtion theory, J. Functionl Anl. No. 119 (1994), 37-78. [3] P.G. Cszz, O. Christensen, nd A.J.E.M. Jnssen, Clssifying tight Weyl- Heisenerg frmes, The Functionl nd Hrmonic Anlysis of Wvelets nd Frmes, Cont. Mth. Vol 247, L. Bgget nd D. Lrson Edts., (1999) 131-148 [4] I. Duechies, The wvelet trnsform, time-frequency locliztion nd signl nlysis. IEEE Trns. Inform. Theory, 36 (5) (199) 961-15. [5] I Duechies, A. Grossmnn, nd Y. Meyer, Pinless nonorthogonl expnsions. J. Mth. Phys. 27 (1986) 1271-1283. [6] I. Duechies, H. Lndu nd Z. Lndu, Gor time-frequency lttices nd the Wexler-Rx identity, J. Fourier Anl. nd Appls. (1) No. 4 (1995) 437-478. [7] R.J. Duffin nd A.C. Scheffer, A clss of non-hrmonic Fourier series. Trns. AMS 72 (1952) 341-366. [8] D. Gor, Theory of communictions. Jour. Inst. Elec. Eng. (London) 93 (1946) 429-457. [9] C. Heil nd D. Wlnut, Continuous nd discrete wvelet trnsforms, SIAM Review, 31 (4) (1989) 628-666. [1] A.J.E.M. Jnssen, Dulity nd iorthogonlity for Weyl-Heisenerg frmes, Jour. Fourier Anl. nd Appl. 1 (4) (1995) 43-436. [11] A.J.E.M. Jnssen, The dulity condition for Weyl-Heisenerg frmes, in Gor Anlysis nd Algorithms: Theory nd Applictions, H.G. Feichtinger nd T. Strohmer Eds., Applied nd Numericl Hrmonic Anlysis, Birkhäuser, Boston (1998) 33-84. [12] I. Reurn nd D. Willims, Morit Equivlence nd Continuous-Trce C - Algers, AMS, Providence, RI, (1998) [13] A. Ron nd Z. Shen, Frmes nd stle sis for shift-invrint suspces of L 2 (R d ), Cndin J. Mth. 47 (1995),151-194 [14] A. Ron nd Z. Shen, Weyl-Heisenerg frmes nd Riesz ses in L 2 (R d ), Duke Mth. J. 89 (1997) 237-282. Dept. of Mthemtics, University of Missouri-Columi, Columi, MO 65211, nd Dept. of Mthemtics, University of South Crolin, Columi, SC 2928 E-mil ddress: pete@mth.missouri.edu;lmmers@mth.sc.edu