On the invertibility of rectangular bi-infinite matrices and applications in time frequency analysis

Transcription

1 On the invertibility of rectangular bi-infinite matrices and alications in time frequency analysis arxiv: v1 [math.ca] 14 May 2007 Götz E. Pfander School of Engineering and Science, Jacobs University Bremen, Bremen, Germany ABSTRACT Finite dimensional matrices having more columns than rows have no left inverses while those having more rows than columns have no right inverses. We give generalizations of these simle facts to bi infinite matrices and use those to obtain density results for frames of time frequency molecules in modulation saces and identifiability results for oerators with bandlimited Kohn Nirenberg symbols. 1. INTRODUCTION Matrices in C m n are not invertible if m n. To generalize this basic fact from linear algebra to bi infinte matrices, we first associate the quadratic shae of M C m n, m = n, to bi-infinite matrices decaying away from their diagonals, more recisely, by matrices M = (m j j) j,j Z with m d j j small for j j large. The rectangular shae of M C m n, m < n, is then taken to corresond to bi-infinite matrices decaying off wedges which are situated between two slanted diagonals of sloe less than one and which are oen to the left and to the right. In short, for λ > 1, we assume m j j small for λ j j ositive andlarge. To this case, we associate the symbol. Similarly, M C m n, m > n, corresonds to bi-infinite matrices that are the adjoints of the matrices described above. That is, the case is described by: for λ < 1, we assume m j j small for λ j + j ositive and large. In both cases, λ 1 corresonds to n 1 in the theory of finite dimensional matrices. m 1

2 We consider bi infinite matrices that act on weighted l saces, 1. To illustrate our main result we first resort to its simlest case. Theorem 1.1. Let M = (m j j) : l 2 (Z) l 2 (Z) and w : R + 0 R + 0 satisfies w(x) = o(x 1 δ ), δ > If m j j < w(λ j j ) for λ j j > 0 and λ > 1, then M has no bounded left inverses. 2. If m j j < w( λ j + j ) for λ j + j > 0 and λ < 1, then M has no bounded right inverses. Note that slanted matrices as covered in [1] and in the wavelets literature [2, 3, 4, 5], decay off slanted diagonals, that is, m j,j small if λj j large. Since λj j λ j j, the results in Section 2 aly in the setting of slanted matrices as well. AfterstatingandrovingourmainresultasTheorem2.1inSection2, weillustrate its usefulness in Section 3 by alying it in the area of time frequency analysis. First, Theorem 2.1 is used to obtain elementary roofs of density theorems for Banach frames of Gabor systems and of time frequency molecules in so-called modulation saces [6, 7]. Second, we discuss how secial cases of Theorem 2.1 have been used to give necessary conditions on the identifiability of seudodifferential oerators which are characterized by a bandlimitation of the oerators Kohn Nirenberg symbols [8, 9, 10]. The background on time frequency analysis that is used throughout Section 3 is given in Section NON INVERTIBILITY OF RECTANGULAR BI-INFINITE MATRICES Let ls (Zd ), 1, s R, be the weighted l -sace with norm {x j } l s = ( )1 {(1+ j ) s x j } l, where {x j } = j x j and {x j } = su j x j. 1 Theorem 2.1. Let 1 1, 2,q 1,q 2, q 1 = 1, q 2 = 1, r 1,r 2,s 1,s 2 R, and M = (m j j) : l 1 s 1 (Z d ) l 2 s 2 (Z d ). 1. If there exists a δ 0 with r 1 s 1 +δ > 0 and d 2 +r 1 +r 2 s 1 +s 2 +δ > 0, and if there exists λ > 1, K 0 > 0, and a function w : R + 0 R+ 0 with w(x) = o (x ) ( )d r q 1 1 r 2 +s 1 s 2 δ 2 and m j j w(λ j j ) (1+ j ) r 1 (1+ j ) r 2, λ j j > K 0, 2

3 then M has no bounded left inverses. 2. If there exists a δ 0 with r 2 s 2 +δ > 0 and d 1 +r 1 +r 2 +s 1 s 2 +δ > 0 and if there exists 0 < λ < 1, K 0 > 0 and a function w : R + 0 R + 0 with w(x) = o (x ) ( )d r 1 q 1 r 2 s 1 +s 2 +δ 2 and m j j w( λ j + j ) (1+ j ) r 1 (1+ j ) r 2, λ j + j > K 0, λ,k 0 > 0, then M has no bounded right inverses. Clearly, Theorem 1.1 is Theorem 2.1 for r 1 = r 2 = s 1 = s 2 = 0, 1 = q 1 = 2 = q 2 = 2, and d = 1. Theorem 2.1 is a direct consequence of Lemma 2.2. Let 1 1,q 1, 2, q 1 = 1, and M = (m j j) : l 1 (Z d ) l 2 (Z d ). ( ) If there exists a function w : R + 0 R+ 0 with w(x) = o x ( )d r q 1 1 r 2 δ 2 satisfying m j j w(λ j j ) (1+ j ) r 1 (1+ j ) r 2, λ j j > K 0, for some constants λ,k 0,r 1,r 2,δ, with λ,k 0 > 1, δ 0, r 1 +δ > 0, and d 2 + r 1 + r 2 +δ > 0, then M has no bounded left inverses. Proof. We begin with the case 1 > 1, 2 < and show that if w : R + 0 R+ 0 satisfies w(x) = o (x ) ( )d r q 1 1 r 2 δ 2, δ 0, r 1 +δ > 0 and d 2 +r 1 + r 2 +δ > 0, then ( ) 2 A K1 = K 2r 1 1 k d 1 w(k) q 1 0 as K 1. (1) K K 1 K 2r2+d 1 We set w(x) = su y x w(y) o v(x)x ( )d r q 1 1 r 2 δ 2. Then ( K K 1 +2 K 2r 2 +d 1 k K k d 1 w(k) q 1 x 2r 2 +d 1 K 1 x 2r 2 +d 1 K 1 k K (x ( 1 q )d r 1 r 2 δ ) and v C 0 (R + ) with w(x) ) 2 K K 1 +1 K 2r 2 +d 1 ( y d 1 w(y) q 1 dy x ( v [K1, ) 2 q 1 2 d+q 1 r 2 +q 1 r 1 +q 1 δ x ( k K+1 ) 2 dx k d 1 w(k) q 1 ) 2 v(y) q 1 y 1 q 1 2 d q 1 r 2 q 1 r 1 q 1 δ dy dx K 1 x 2r 2 +d 1 x d 2r 2 2 r 1 2 δ dx ) 2 v [K1, ) 2 (r 1 +δ)(q 1 d+ 2 q 1 r q 1 r q 1 δ) K 2r 1 2 δ 1 = o(k 2r 1 1 ), 3

4 since v [K1, ) 0 as K 1 and (1) follows. To show that inf x l0 (Z ){ Mx d l 2 x l 1 } = 0, we fix ǫ > 0 and note that (1) rovides us with a K 1 > K 0 satisfying A K1 (2 d d) 2 1 ( 2 2r 2 λ 1 ) 2 r 1ǫ 2. λ Set N = λ(k1 +1) λ 1 and Ñ = N + K λ 1. Then λ(k 1+1) N λ(k 1+2) λ 1 λ 1 imlies λn λk 1 +λ+n and N K 1 + N λ +1 > K 1 + N λ = Ñ. Therefore, (2Ñ +1)d < (2N + 1) d and the matrix M = (m j j) j e N, j N : C(2N+1)d C (2 e N+1) d has a nontrivial kernel. We now choose x C (2N+1)d with x 1 = 1 and M x = 0 and define x l 0 (Z 2 ) according to x j = x j if j N and x j = 0 otherwise. By construction, we have x l 1 = 1, and (Mx) j = 0 for j Ñ. To estimate (Mx) j for j > Ñ, we fix K > K 1 and one of the 2d(2 ( N + K) ) d 1 indices λ j Z d with j = N +K. We have λ λj N +Kλ and λ j j Kλ K for all j Z d with j N. Therefore (Mx) j q 1 = j N (1+ j ) q 1r 2 m j jx j q 1 x q 1 1 j N (1+ j ) q 1r 2 (N+1) q 1r 1 j N m j j q 1 (1+ j ) q 1r 1 w(λ j j ) q 1 j K w( j ) q 1 = 2 d d(1+ j ) q 1r 2 (N+1) q 1r 1 k d 1 w(k) q 1. 4 k K

5 Finally, we comute Mx 2 l 2 = (Mx) j 2 = j Z d (2 d d) 2 q 1 j N λ +K 1 (2 d d) 2 q 1 (N+1) 2r 1 j N λ +K 1 (Mx) j 2 (1+ j ) 2r 2 (N+1) 2r 1 k d 1 w(k) k j k d 1 w(k) q 1 K N λ 12d(2K) d 1 (K +1) 2r2( +K k K ( k d 1 w(k) q 1 ( ) 2 (2 d d) 2 q +1 r 1 2 λ(k1 2r 2 +2) 1 +1 K 2r 2 +d 1 λ 1 K N λ +K 1 ( ) ( 2 (2 d d) r 2 q r 2 λ 1 (K 1 +3) 2r 1 K 2r 2 +d 1 k d 1 w(k) q 1 λ 1 K N λ +K 1 k K ǫ 2, that is, Mx l 2 ǫ. Since ǫ was chosen arbitrarily and x l 1 = 1, we have inf x l0 (Z 2 ){ Mx l 2 x l 1 } = 0 and M is not bounded below and has no bounded left inverses. The cases 1 = 1 and/or 2 = follow similarly. Proof of Theorem 2.1. Part 1. Let M = (m j j) : l 1 s 1 (Z d ) l 2 s 2 (Z d ) satisfy the hyothesis of Theorem 2.1, art 1. Suose that, nonetheless, M = (m j j) : l 1 s 1 (Z d ) l 2 s 2 (Z d ) has a bounded left inverse. This clearly imlies that k K 2 ) 2 M = ( m j j) = ( m j j (1+ j ) s 2 (1+ j ) s 1 ) : l 1 (Z d ) l 2 (Z d ) has a bounded left inverse which contradicts Theorem 2.2, since for λ j j > K 0, we have m j j = mj j(1+ j ) s 2 (1+ j ) s 1 ) w(λ j j ) (1+ j ) r 1 s 1 (1+ j ) r 2+s 2 d with δ 0, r 1 s 1 +δ > 0, 2 +r 1 +r 2 s 1 +s 2 +δ > 0, and w(x) = o (x ) ( )d r q 1 1 r 2 +s 1 s 2 δ 2. 5 ) 2 ) 2

6 Part 2. The matrix M : l 1 s 1 (Z d ) l 2 s 2 (Z d ) has a bounded right inverse if and only if its adjoint M : l 2 s 2 (Z d ) l 1 s 1 (Z d ) has a bounded left inverse. The conditions on M in Theorem 2.1, art 2 are equivalent to the conditions on M in Theorem 2.1, art 1. The result follows. 3. APPLICATIONS Before stating alications of Theorem 2.1, we give a brief account of the concets from time frequency analysis that aear in this section. For additional background on time frequency analysis and, in articular, Gabor frames, see [11] Time frequency analysis and Gabor frames The Fourier transform of a function f L 1 (R d ), is given by f(γ) = f(x)e 2πix γ dx, γ R d, where R d is the dual grou of R d, and which, aside of notation, equals R d. The Fourier transform can be extended to act unitarily on L 2 (R d ) and isomorhically on the dual sace of Schwarz class functions S(R d ), that is, on the sace of temered distributions S (R d ) S(R d ). The translation oerators T y : S(R d ) S(R d ), y R d, is given by (T y f)x = f(x y), x R d, and the modulation oerator M ξ : S(R d ) S(R d ) is given by (M ξ f)x = e 2πixξ f(x), x R d. Both extend isomorhically to S (R d ), and so do their comositions, the so-called time frequency shifts π(z) = π(y,ξ) = T y M ξ, z = (y,ξ) R d R d. Note that theadjoint oerator π(z) of π(z) = π(y,ξ)isπ(z) = e 2πiyξ π( z). The short time Fourier transform V g f of f L 2 (R d ) S (R d ) with resect to a window function g L 2 (R d )\{0} is V g f(z) = f,π(z)g = f(x)g(x y)e 2πi(x y) ξ dx, z = (y,ξ) R d R d. R d We have V g f L 2 (R d R d ) and V g f L 2 = f L 2 g L 2. A central goal in Gabor analysis is to find g L 2 (R d ) and full rank lattices Λ = AZ 2d R d R d, A R 2d 2d full rank, which allow the discretization of the formula V g f L 2 = f L 2 g L 2 in the following sense: for which g L 2 (R d ) and full rank lattices Λ exists A,B > 0 with A f 2 L 2 z Λ V g f(z) 2 B f 2 L 2, f L2 (R d ). (2) If (2) is satisfied, then (g,λ) = {π(z)g} z Λ is called Gabor frame for the Hilbert sace L 2 (R d ). More recently, the question above has been considered for general sequences Γ in R d R d in lace of full rank lattice Λ [12, 13, 14]. 6

7 To generalize (2) to Banach saces, we adot the definition of -frames from [15]. Definition 3.1. The Banach sace valued sequence {g j } j Z d X, d N, is an ls frame for the Banach sace X, 1, s R, if the analysis oerator C F : X ls (Zd ), f { f,g j } j is bounded and bounded below, that is, if there exists A,B > 0 with A f X { f,g j } l s B f X, f X. (3) Note that in the Hilbert sace case X = L 2 (R d ) and l s(z 2d ) = l 2 (Z 2d ), (2) imlies that C F has a bounded left inverse, while in the Banach sace case (3) does not rovide us with a left inverse. Therefore, the existence of a bounded left inverse for C F is included in the definition of the standard generalization of frames to Banach saces [16, 17, 18]. Analogously to Definition 3.1, we include a generalization of Riesz bases in the Banach sace setting. Definition 3.2. A sequence {g j } j Z d X, d N is called l s Riesz basis in the Banach sace X, 1, s R, if the synthesis oerator D {gj } j : l s (Z2d ) X, {c j } j j c jg j is bounded and bounded below, that is, if there is A,B > 0 with A {c j } j l s j c j g j X B {c j } j l s, {c j } j l s (Zd ). The Banach saces of interest here are the so called modulation saces[19, 20, 21]. Clearly, V g f(z) = f,π(z)g, z R d R d is well defined whenever g S(R d ) and f S (R d )(orviceversa). Thistogetherwith V g f L 2 = g L 2 f L 2 inthel 2 theory motivates the following. We let g = g S(R d ) be an L 2 normalized Gaussian, that is, g(x) = 2 d 4 e π x 2 2, x R d, and define the modulation sace M s (Rd ), s R, 1, by M s(r d ) = {f S (R d ) : V g f L s(r d R d )} with Banach sace norm f M s = V g f L s = ( (1+ z ) s V g f(z) dz and the usual adjustment for =. )1 <, 1 <, Examle 3.3. For λ < 1, (g, λz 2d ) is an l 2 frame for L 2 (R d ) [22, 23]. Since g S(R d ) M 1 t (Rd ) for all t 0, Theorem 20 in [14] imlies that in this case (g, λz 2d ) is an l s frames for M s (Rd ) for s R and 1. The Wexler-Raz 7

8 identity imlies that for λ > 1, (g, λz 2d ) is an l 2 Riesz basis in L 2 (R d ). Hence, D (g,λz 2d ) : l 2 (Z 2d ) L 2 (R d ) has a bounded left inverse of the form C (eg,λz 2d ) where the so called dual function g of g satisfies g S(R d ) [24]. The oerator C (eg,λz 2d ) is a bounded oerator maing M s(r d ) to l s(z 2d ). This imlies that D (g,λz 2d ) has a left inverse and (g, λz 2d ) is an l s Riesz basis in M s (Rd ) for s R and Density results for Gabor ls frames in modulation saces One of the central results in Gabor analysis is the fact that (g,λ), g L 2 (R d ), cannot be a frame for L 2 (R d ) if the measure of a fundamental domain of the full rank lattice Λ is larger than 1 [25, 26, 27]. Generalizations of this result to general sequences Γ in R d R d require an alternative definition of density [12, 28, 29]. Definition 3.4. Let Q R = [ R,R] 2d R d R d and let Γ be a sequence of oints in R d R d. Then D (Γ) = liminf R Γ Q R +z inf z R d R bd (2R) 2d and D + (Γ) = limsu R Γ Q R +z su z R d R b (2R) 2d d are called lower and uer Beurling density of Γ. If D + (Γ) = D (Γ), then Γ is said to have uniform density D(Γ) = D + (Γ) = D (Γ). Remark 3.5. The density of a sequence Γ does not equal the density of its range set. For examle, the density of the sequence {..., 2, 2, 1, 1,0,0,1,1,2,2,3,3,...} in R is 2, while the density of the range of the sequence, namely of Z, is 1. In [30], it was shown that if (g,γ), g L 2 (R d ), Γ R d R d, is an l 2 frame for L 2 (R d ) = M 2 0 (Rd ), then 1 D (Γ) D + (Γ) <, a result that has recently been refined by Theorem 3 and Theorem 5 in [13]. For l s frames for M s(r d ), Theorem 2.1 imlies Theorem 3.6. Let 1, s R, and g M2d if s < 0 and and g M2d+δ, δ > s,0 else. If (g,γ) is an l s frame for Ms(R d ), then D + (Γ) 1. Proof. Let Γ be given with D + (Γ) < 1. We choose λ > 1 with 1 > λ 4d > D + (Γ) and R 0 > 0 with Γ Q R < su z R d b R d Γ Q R +z < λ 4d (2R) 2d, R > R 0. Since D + (Γ) <, the sequence Γ has no accumulation oints and we can enumerate thesequence ΓbyZ 2d sothat γ j γ j imlies j j forj,j Z 2d. This gives, γ j / Q R if (2 j 1) 2d = (2( j 1)+1) 2d λ 4d (2R) 2d, R > R 0, 8

9 D (g,λz 2d ) M s (Rd ) l s (Z2d ) C (g,γ) ls (Z2d ) M cj π(λj)g D (g,λz 2d ) {c j } j C (g,γ) { c j π(λj),π(γ j )g } j M Figure 1. Sketch of the roof of Theorem 3.6. We choose λ > 1 so that (g, λz 2d ) is an l s Riesz basis in M s(r d ), so D (g,λz 2d ) is bounded below. Theorem 2.1 alies to M = C (g,γ) D (g,λz 2d ), showing that M is not bounded below. This imlies that C (g,γ) is not bounded below and has no bounded left inverses. and, therefore, γ j / Q λ 2 j λ2 2 for λ 2 j λ2 2 > R 0. (4) We have { } C (g,γ) D (g,λz 2d ) : ls (Z2d ) ls (Z2d ), {c j } j c j π(λj)h,π(γ j )g = M{c j } j, with M = (m j j) and m j j = π(λj)h,π(γ j )g = V g h(γ j λj). and so where Note that (4) imlies γ j λj λ 2 j λ2 2 λj = λ ( λ j j λ 2), m j j = π(λj)g,π(γ j )g = V g g(γ j λj) w(λ j j ) ( w( z ) = (1+ z ) 2d δ su (1+ z ) 2d+δ V g g( z) ), z R d R d. ez A direct alication of Theorem 2.1 imlies that C (g,γ) D (g,λz 2d ) is not bounded below. Since D (g,λz 2d ) is bounded below, we conclude that C (g,γ) is not bounded below which comletes the roof. 9 j

10 Note that the last lines in the roof of Theorem 3.6 can be modified to aly to time frequency molecules which we shall consider in the following. We say that a sequence {g j } j of functions consist of at Γ = {γ j } j (v,r 1,r 2 ) localized time frequency molecules if V g g j (z) (1+ z ) r 1 (1+ j ) r 2 w( z γ j ), w = o(x v ). (5) If (5) is satisfied for r 1 = r 2 = 0, then we simly seak of at Γ v localized time frequency molecules. Note that if {g j } j (M s (Rd )) is (v,r 1,r 2 ) localized, then by definition {g j } j M v r 1 (R d ), and, consequently, if v r 1 > 2d we have {g j } j M 1 (R d ), a fact which we take into consideration when stating the hyothesis of Theorem 3.7 and Theorem 3.8 Related concets of localization were introduced in [1, 14, 12, 13], artly to obtain density results and artly to describe the time frequency localization of dual frames of irregular Gabor frames (see also Remark 3.10). Theorem 3.7. If {g j } j (M s (Rd )) M v r 1, 1, s R is an l s frame for M s(r d ) which is (v,r 1,r 2 ) localized at Γ = {γ j } j, with δ s, v r 1 r 2 2d δ, r 1 + 2d +δ > 0 and δ 0, then D+ (Γ) 1. Note that Theorem 9 in [13] states that if {g j } is an l 2 frame for L 2 (R d ) which consists of at Γ d +δ localized time frequency molecules, δ > 0, then actually 1 D (Γ). Below, we show that comonents of the roof of Theorem 2.2 can be used to obtain some of the density results given above with D + (Γ) being relaced by D (Γ). Theorem 3.8. If {g j } j M 1 (R d ) is an l frame for M (R d ), 1, which is 2d+δ localized at Γ = {γ j } j with D + (Γ) < and δ > 0, then D (Γ) 1. Proof. Suose that {g j } j is an ls frame for M (R d ) which is 2d+δ localized at Γ = {γ j } j, D (Γ) < 1. For z 0, α 3 chosen below, we shall consider the Gabor system {π(α3 1 j+z 0 )g} j Z 2d which is an l Riesz basis for M (R d ). We shall show that {g j } is not an l frame by arguing that C {gj } D {π(α 1 3 inf j+z 0)g} x l x l (Z d ) x l To this end, fix ǫ > 0. We first assume 1 < <. Since D + (Γ) <, there exists α 1 1 and R 0 1 with > α 2d 1 > D + (Γ) 0 and Γ Q R +z α 2d 1 (2R)2d, z R d R d, R R = 0.

11 Further, we can ick α 2,α 3 > 1 2 with D (Γ) < α 2d 2 < α 2d 3 < 1, and N with α 2 +α 1 ( ( 1+ 1 ) 2d 1 ) 2d < α 3 (1 1 2 ) 2d. We now choose a monotonically decreasing w(x) = o(x 2d δ ) with V g g j (z) w( z γ j ). As demonstrated in the roof of Theorem 2.2, w = o(x 2d δ ), δ > 0, allows us to ick K 2 such that for all K 2 K 2 (2 2d 2d) q +1 K K 2 K 2d 1 k α 3 2α 1 K k 2d 1 w(k) q q < ǫ. Also, there exists R 0, N 0 = α 3 R 0, such that there exists z 0 R d R d with Q R0 +z 0 Γ α 2d 2 (2R 0) 2d ; R 0 R 0 ; N 0, α 1 α 2 R0 ; (5 α 1 α 3 R 0 ) 2d w ( ) R 0 2 < ǫ; K 1 = N 0 1 α 2 N 0 > 1; K 2 = 2 ( ) α 1 α 3 N 0 α 2 N 0 K 2,K 1. The sequence Γ has no accumulation oint since D + (Γ) < which imlies that we can choose an enumeration of the sequence Γ by Z 2d with j j if γ j z 0 γ j z 0, j,j Z 2d. As mentioned earlier, we set g j = π ( α3 1 j +z 0) g for j Z 2d, and M = (m j j) = ( g j,g j ). The matrix M = (m j j) j N 0 1, j N 0 : C (2N 0+1) d C (2N 0 1) d has a nontrivial kernel, so we may choose x C (2N 0+1) d with x = 1 and M x = 0 and define x l 0 (Z 2 ) according to x j = x j if j N 0 and x j = 0 otherwise. To estimate the contributions of (Mx) j for j Z 2d to Mx l, we consider three cases. Case 1. j α 2 N 0 +K 1 = N 0 1. This imlies (Mx) j = 0 by construction. 11

12 Case 2. α 2 N 0 +K 1 < j α 2 N 0 +K 2. Observe that the set Q R0 + R 0+z 0 \ Q R0 +z 0 consists of a finite number of hyercubes of width R 0 R 0, so we can estimate Q R0 + R 0+z 0 Γ α2 2d (2R 0 ) 2d +α1 2d ( (2R 0 ) 2d α2 2d +α2d 1 ( ( 2( ( ) 2d (2α3 1 N 0) 2d α3 2d ( ) 2d 2N 0 2N 0 2 (2N0 1) 2d )) ) 2d R 0 + R 0 (2R0 ) 2d ( ( ) )) 2d Hence, for any j with j N 0 = α 2 N 0 +K 1 +1, we have γ j / Q R 0 + R 0 +z 0 and, therefore, for j N 0 = α 3 R 0 we have α 1 3 j+z 0 γ j = (γ j z 0 ) α 1 3 j R 0 + R 0 α 1 3 α 3R 0 R 0 α 1 3 R 0 2, and, therefore, m j j = g j,g j = V g g j (α3 1 j +z 0) w ( α3 1 j +z ) ( ) 0 γ j w R 0 2. This gives Mx {j : α 2 N 0 +K 1 < j α 2 N 0 +K 2 } = w w w α 2 N 0 +K 1 < j α 2 N 0 +K 2 α 2 N 0 +K 1 < j α 2 N 0 +K 2 ( ) R 0 2 ( R 0 α 2 N 0 +K 1 < j α 2 N 0 +K 2 m j jx j j N 0 m j j q j N 0 (2N 0 +1) 2d q q j N 0 x xj ) 2 (2 2 α 1 α 3 N 0 +1) 2d (2N 0 +1) 2d q ( ) R (5 0 α 2 1 α 3 R 0 ) 2d(1+ q ) ǫ (6) Case 3. α 2 N 0 + K 2 < j. For such j, we set N = j and obtain α1 1 (N 1 ) 2 α 1 1 ( α 2 N 0 +K ) α 2 2 α 1 N 0 R 0, and, hence, Γ Q α 1 1 (N 1 2 ) +z 0 α 2d 1 (2α 1 1 (N 1 2 ))2d = (2N 1) 2d. 12

13 This imlies γ j / Q α 1 1 ( j 1 2 ) + z 0. Similarly as in Case 2., we fix j, K with j = α 2 N 0 +K, K > K 2, and conclude that for j N 0, α 1 3 j +z 0 γ j = (γ j z 0 ) α 1 3 j α 1 1 ( j 1 2 ) α 1 3 j Therefore, (Mx) j q = α 3 j j α 3 α 1 2α 1 α 3 α 2 N 0 +2 α 3 α 1 α 3 2α 1 ( K 2 K N 0 α 3 2α 1 2α ( ) 1 ) α1 N 0 α 2 N 0 1 α 3 + α 3 2α 1 K α 3 2α 1 K. m j jx j q x q m j j q j N 0 j N 0 ( α3 w j j α ) q 3 α 1 2α 1 j N 0 w( j ) q = 2(2d)(2k) 2d 1 w(k) q j α 3 K 2α 1 = 2 2d 2d k α 3 2α 1 K k 2d 1 w(k) q. k α 3 2α 1 K Finally, we comute q (Mx) j (2 2d 2d) q k 2d 1 w(k) q j > α 2 N 0 +K 2 j α 2 N 0 +K 2 k α 3 j 2α 1 (2 2d 2d) q 2(2d)(2K) 2d 1 k 2d 1 w(k) q K α 2 N 0 +K 2 (2 2d 2d) q +1 K α 2 N 0 +K 2 K 2d 1 k α 3 2α 1 K k α 3 2α 1 K k 2d 1 w(k) q 2 q q ǫ (7) by hyothesis. Clearly, (6) and (7) give Mx l 2 1 ǫ which comletes the roof for 1 < <. The cases = 1 and = follow similarly. Remark 3.9. If {g j } = (g,γ) and the analysis oerator C (g,γ) is bounded, then D + (Γ) < follows [30]. If {g j } are only assumed to be Γ localized time frequency 13

14 molecules, then boundedness of C {gj } does not imly D + (Γ) <. For examle, consider {g j } = { 1 k! g} k N. Remark Theorem 9 in [13] imlies that time frequency molecules {g j } which are v localized at Γ = {γ j }, v > d, and which generate an l 2 frame for L 2 (R) satisfy 1 D (Γ) D + (Γ). Further, Theorem 22 in [14] states that under the same hyothesis but v > 2d+s imlies that being an l 2 frame for L 2 (R d ) is equivalent to being an l s frame for M s (Rd ) for all 1 and all s 0. This result alone does not imly Theorem 3.7 nor Theorem 3.8 as they only assume that {g j } is an l s frame for M s(r d ) for some and s. Under stronger conditions, [1] fills this ga. Namely, Theorem 3.1 and Examle 3.1 in [1] show that if v > (2d+1) 2 +2d and {g j } is an at Γ = {γ j } v localized l frame for M (R d ) for one, 1, then {g j } is an l frame for M (R d ) for all and therefore for the well studied case = 2 [13]. This imlies Theorem 3.8 for v > (2d+1) 2 +2d Identification of oerators with bandlimited Kohn Nirenberg symbols A central goal in alied sciences is to identify a artially known oerators H from a single inut outut air (g,hg). We refer to an oerator class H as identifiable, if there exists an element g in the domain of all H H that induces a ma Φ g : H Y, H Hg which is bounded and bounded below as ma between Banach saces. In [8, 9], secial cases of Theorem 2.1 layed a crucial role in showing that classes of seudodifferential oerators with Kohn Nirenberg symbol bandlimited to a rectangular domain [ a, a 2 2 ] [ b, b ] are not identifiable if ab > 1. The bandlimitation of 2 2 a Kohn Nirenberg symbol to a rectangular domain [ a, a 2 2 ] [ b, b ] can be exressed 2 2 by a corresonding suort condition on the oerators so-called sreading function η 1 H. Consequently, we consider oerators H : D Ms (R), D M (R), included in } Hs {H ([ a, a 2 2 ] [ b, b]) = = η 2 2 H (z)π(z)dz, η H M [ a 2,a 2 ] [ b 2,b 2 ] s (R R) (8) and with norm H H s = η H M s. The integral in (8) is defined weakly using Hf,h = η H,V h f 2 [9]. In [8] it was shown that Theorem There exists g M (R) with Φ g : H 2 0 ([ a 2, a 2 ] [ b 2, b 2 ]) M2 0 (R) bounded and bounded below if and only if ab 1. 1 Infact, thesreadingfunctionofanoeratoristhesymlecticfouriertransformoftheoerator s Kohn Nirenberg symbol [8, 10]. 2 Here,, is taken to belinear in the first comonent and conjugate linear in the second. 14

15 Hs (R) Φ g Ms (R) j c Φ g jp j j c jp j g D {Pj } C (g,λz 2d ) D {Pj } C (g,λz 2d ) l s (Z2 ) M l s (Z2 ) {c j } j M { c j P j g,π(λj )g} j Figure 2. Sketch of the roof of Theorem We choose a structured oerator family {P j } H s so that the corresonding synthesis ma D {Pj } : {c j } c j P j has a bounded left inverse. Further, C (g,λz 2d ) has a bounded left inverse for λ < 1. We then use Theorem 2.1 to show that for any g M (R), the comosition M = C (g,λz 2d ) φ g D {Pj } is not bounded below, therefore imlying that φ g : H s M s (R) is not bounded below as well. Note that H0 1([ a, a 2 2 ] [ b, b ]) consists of Hilbert Schmidt oerators, the norm 2 2 H 2 0 is equivalent to the Hilbert Schmidt sace norm, and M 2 0 is a scalar multile of the L 2 norm. The main result in [9] is Theorem For ab < 1 exists g M (R) with Φ g : H0 ([ a, a 2 2 ] [ b, b ]) 2 2 M0 (R) bounded and bounded below, while for ab > 1 exists no such g M (R). Here, we use the generality of Theorem 2.1 to obtain Theorem Let 1 and s R. For ab > 1 exists no g M (R) with Φ g : Hs ([ a, a 2 2 ] [ b, b]) 2 2 M s (R) bounded and bounded below. Sketch of roof. We assume a = b and a 2 > 1. The general case ab > 1 follows similarly. The goal is to show that for any g M (R) which induces a bounded oerator Φ g : H s([ a 2, a 2 ]2 ) M s(r), this oerator is not bounded below. To see this, we ick λ > 1 with 1 < λ 4 < a 2 and define a rototye oerator P H s ([ a 2, a 2 ]2 ) via its sreading function η P (t,ν) = η(t)η(ν) where η is smooth, takes values in [0,1] and satisfies η(t) = 1 for t a/2 a/2λ and η(t) = 0 for t a/2 a/2. The collection of functions {Mλ a jη P} j Z 2 corresonds to the oerator family {π( λ a j)pπ(λ a j) } j Z 2 [9]. Further, it forms a Riesz basis for its closed linear san 15

16 in L 2 (R R) and, for c > 0 sufficiently large, the collection {π( λ a j, 1 c k)η P} j,k Z 2 is a frame for L 2 (R 2 ) [11, 31]. Arguing as in Examle 3.3, we obtain a bounded left inverse of D {Mλa j η P} : l s (Z2 ) M s (R R), thereby showing that D {Mλa j η P} and also the corresonding oerator synthesis ma D {Pj } : l s (Z2 ) H s (R R) with P j = π( λ a j)pπ(λ a j), j Z 2, are bounded below. M s For any fixed g M (R) which induces a bounded ma Φ g : Hs([ a, a 2 2 ]2 ) (R) we consider the oerator M = (m jj ) = C (g, λ 2 ) Φ g D {Pj }: ls (Z2 ) ls (Z2 ). a We have mjj = π( λ a j)pπ(λ a j) g, π( λ2 a j )g = Vg Pπ( λ a j) g ( λ a (λj j) ). In [8] it is shown that smoothness and comact suort of η P imlies that there exist nonnegative functions d 1 and d 2 on R, decaying raidly at infinity, such that for all g M (R), Pg(x) g M d 1 (x) and Pg(ξ) g M d 2 (ξ). This imlies that V g Pπ( λ a j) g decays raidly and indeendently of j, so that we can aly Theorem 2.1 to show that M is not bounded below. Since λ2 < 1, Examle 3.3 imlies that C a (g, λ2 a ) isboundedbelow. Also, D {Pj } isboundedbelow, imlying thatφ g cannot bebounded below. Since g M (R) was chosen arbitrarily, this comletes the roof. References [1] A. Aldroubi, A. Baskakov, I. Krishtal, Slanted matrices, banach frames, and samling, Prerint (2007). [2] A. Cavaretta, W. Dahmen, C. Micchelli, Stationary subdivision, Vol. 93, [3] W. Dahmen, C. Micchelli, Banded matrices with banded inverses. II. Locally finite decomosition of sline saces, Constr. Arox. 9 (2-3) (1993) [4] M. Gasca, C. A. Micchelli, J. M. Peña, Banded matrices with banded inverses. III. -slanted matrices, in: Wavelets, images, and surface fitting (Chamonix-Mont-Blanc, 1993), A K Peters, Wellesley, MA, 1994, [5] C. Micchelli, Banded matrices with banded inverses, J. Comut. Al. Math. 41 (3) (1992) [6] H. Feichtinger, K. Gröchenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (2) (1997) [7] H. Feichtinger, Atomic characterizations of modulation saces through Gabor-tye reresentations, in: Proc. Conf. Constructive Function Theory, Edmonton, July 1986, 1989, [8] W. Kozek, G. Pfander, Identification of oerators with bandlimited symbols, SIAM J. Math. Anal. 37 (3) (2006) [9] G. Pfander, D. Walnut, Oerator identifcation and Feichtinger s algebra, Saml. Theory Signal Image Process. 5 (2) (2006) [10] G. Pfander, D. Walnut, Measurement of time variant channels, IEEE Trans. Info. Theory 52 (11) (2006)

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