PERTURBATIONS OF BANACH FRAMES. Abstract. Banach frames and atomic decompositions are sequences which have basis-like properties

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1 PERTURBATIONS OF BANACH FRAMES AND ATOMIC DECOMPOSITIONS OLE CHRISTENSEN y an CHRISTOPHER HEIL z Abstract. Banach frames an atomic ecompositions are sequences which have basis-like properties but which nee not be bases. In particular, they allow elements of a Banach space to be written as combinations of the frame or atomic ecomposition elements in a stable manner. However, these representations nee not be unique. Such exibility is important in many applications. In this paper, we prove that frames an atomic ecompositions in Banach spaces are stable uner small perturbations. Our results are strongly relate to classic results on perturbations of Paley/Wiener an Kato. We also consier uality properties for atomic ecompositions, an iscuss the consequences for Hilbert frames. Key wors. atomic ecompositions, Banach frames, frames, perturbations AMS(MOS) subject classications. 42C99, 46B99, 46C99 1. Introuction. Frames for Hilbert spaces were introuce by Dun an Schaeer [DS] as part of their seminal research in nonharmonic Fourier series. Daubechies, Grossmann, an Meyer [DGM] later foun a funamental new application, to wavelet an winowe Fourier transforms. Frames continue to play an important role in each of these areas. A set of vectors fy i g in a Hilbert space H is a (Hilbert) frame if the norms kxk H an kfhx; y i igk`2 are equivalent. Dene Ux = fhx; y i ig. Then U Ux = P hx; y i i y i is an invertible mapping of H onto itself. With ~y i = (U U)?1 y i, we have the reconstruction formulas x = hx; ~y i i y i = hx; y i i ~y i : (1) The collection f~y i g also forms a frame, the ual frame of fy i g. The representations in (1) nee not be unique: fy i g nee not be a basis. A frame which is a basis must be a Riesz basis. Conversely, all Riesz bases are frames. The basic theory of frames in Hilbert spaces can be foun in Dun an Schaeer's original paper [DS], Young's classic text [Y], Daubechies' paper [D1] an book [D2], or the research-tutorial [HW]. Frames were extene to Banach spaces by Grochenig [G]. In Hilbert spaces, it is a remarkable fact that the norm equivalence hypothesis leas to the reconstruction fory Institut fur Mathematik, Universitat fur Boenkultur, Gregor Menel{Strasse 33, A-118 Wien, Austria (olechr@pap.univie.ac.at). z School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia USA an The MITRE Corporation, Befor, Massachusetts 173 USA (heil@math.gatech.eu). This author was partially supporte by National Science Founation Grant DMS an the MITRE Sponsore Research Program. 1

2 2 ole christensen an christopher heil mula (1). This oes not hol in Banach spaces in general. A ecomposition of a Banach space is therefore ene as follows. Definition 1. Let be a Banach space an let be an associate Banach space of scalar-value sequences inexe by N = f1; 2; 3; : : :g. Let fy i g i2n an fx i g i2n be given. If: (a) fhx; y i ig 2 for each x 2, (b) the norms kxk an kfhx; y i igk are equivalent, an (c) x = P 1 i=1 hx; y ii x i for each x 2, then (fy i g; fx i g) is a (linear) atomic ecomposition of with respect to. If the norm equivalence is given by A kxk kfhx; y i igk B kxk, then A, B are a choice of atomic bouns for (fy i g; fx i g). Some examples of atomic ecompositions in Banach function spaces are the \transform" of Frazier an Jawerth [FJ] an Feichtinger an Grochenig's constructions [FG], [G]. An atomic ecomposition provies a factorization of the ientity map I on. That is, I is written as a composition of the coecient mapping x 7! fhx; y i ig an the reconstruction operator fc i g 7! P c i x i. Such series reconstructions are theoretically appealing. However, for numerical implementations it is often preferable to formulate the reconstruction operator via an iteration or other algorithm. We therefore make the following enition, which allows freeom in the form of the reconstruction operator. Definition 2. Let be a Banach space an let be an associate Banach space of scalar-value sequences inexe by N. Let fy i g i2n an S:! be given. If: (a) fhx; y i ig 2 for each x 2, (b) The norms kxk an kfhx; y i igk are equivalent, an (c) S is boune an linear, an Sfhx; y i ig = x for each x 2, then (fy i g; S) is a Banach frame for with respect to. The mapping S is the reconstruction operator. If the norm equivalence is given by A kxk kfhx; y i igk B kxk, then A, B are a choice of frame bouns for (fy i g; S). Note that if U:! is the coecient mapping ene by Ux = fhx; y i ig then

3 perturbations of banach frames 3 ksk?1, kuk are a choice of frame bouns for the Banach frame (fy i g; S). Our purpose in this note is to prove that atomic ecompositions an Banach frames are stable uner small perturbations. This is inspire by corresponing classical perturbation results, e.g., the Paley-Wiener basis stability criteria [PW], [Y] an the perturbation theorem of Kato [K]. We introuce new an weaker conitions which still ensure the esire stability. This is not only of theoretical interest: we show that our results can be applie to coherent state frames. This is not the case if stanar basis perturbation results are merely generalize irectly to frames. (Retherfor an Holub [RH] is an excellent survey of perturbation results for bases.) In aition, we investigate uality properties of atomic ecompositions, an consier the consequences of our results for Hilbert frames. Relate Hilbert space results, partially inspire by a weak version of Theorem 2 rst prove in [He], have appeare in [C1], [C2], [C3]. We assume throughout that sequences are inexe by N. We use the following terminology. We say that a sequence fc i g is nite if only nitely many components are nonzero. A Banach space of sequences is soli if whenever fb i g an fc i g are sequences with fc i g 2 an jb i j jc i j then it follows that fb i g 2 an kfb i gk kfc i gk. For example, let e j enote the elta sequence e j (i) = ij. If fe j g j2n forms an unconitional basis for then is soli. It also follows in this case that has an absolutely continuous norm, i.e., if fc i g 2 then lim n!1 kfc i? c i I n (i)gk =, where fi n g is any family of subsets of N such that I 1 I 2 % N an In is the characteristic function In (i) = 1 if i 2 I n, if i =2 I n. In particular, the hypotheses on, in most of our results are satise if can be realize as a Banach space of sequences of scalars an if fe j g forms an unconitional basis for both an. 2. Perturbation results. We rst show that Banach frames are stable uner small perturbations of the frame elements. Theorem 1. Let (fy i g; S) be a Banach frame for with respect to. Let fz i g. If there exist, such that (a) kuk + < ksk?1, an (b) kfhx; y i? z i igk kfhx; y i igk + kxk for all x 2, then there exists a reconstruction operator T such that (fz i g; T ) is a Banach frame for

4 4 ole christensen an christopher heil with respect to with frame bouns ksk?1? ( kuk + ), kuk + ( kuk + ), where U is the coecient mapping Ux = fhx; y i ig. Proof. The hypotheses imply that the operator V :! ene by V x = fhx; z i ig is boune an satises for all x 2. Therefore, kux? V xk kuxk + kxk kv xk (kuk + kuk + ) kxk : This establishes the upper frame boun. For the lower boun, observe that SU = I, so ki? SV k ksk ku? V k ksk ( kuk + ) < 1: Therefore SV is invertible, an k(sv )?1 k (1? ( kuk + ) ksk)?1. Finally, if we set T = (SV )?1 S then T V = I, an kxk kt k kv xk ksk 1? ( kuk + ) ksk kv xk : This gives the esire lower boun: 1 ksk? ( kuk + ) kxk kv xk : The hypotheses in Theorem 1 are natural from the point of view of perturbation of operators: they mean that the operator U? V is relatively boune with respect to U [K, p. 181]. In Section 4 we apply this result to Hilbert frames. For atomic ecompositions, we can perturb in instea of. Our result is a \Paley- Wiener Theorem for atomic ecompositions" [Y, p. 38]. Theorem 2. Suppose that has an absolutely continuous norm. Let (fy i g; fx i g) be an atomic ecomposition of with respect to with bouns A, B. Let fw i g. If there exist, such that (a) + B < 1, an (b) P c i (x i?w i ) P c i x i + kfc i gk for any nite sequence fc i g 2, then there exists a family fz i g such that (fz i g; fw i g) is an atomic ecomposition of with respect to with bouns A (1 + ( + B))?1, B (1? ( + B))?1. Moreover,

5 perturbations of banach frames 5 fw i g is a basis for if an only if fx i g is a basis for. Proof. Because of the assumption that has an absolutely continuous norm, the series P hx; y i i w i is convergent for any x 2. If we ene T :! by T x = P hx; y i i w i, then kx? T xk kxk + kfhx; y i igk ( + B) kxk for all x 2. Therefore ki? T k < 1, so T is invertible. Dene z i = (T?1 ) y i. Then Further, x = T T?1 x = ht?1 x; y i i w i = hx; z i i w i : A kt k kxk A kt?1 xk kfht?1 x; y i igk B kt?1 xk B kt?1 k kxk ; so (fz i g; fw i g) is an atomic ecomposition of with respect to. Since kt k 1++ B an kt?1 k (1? ( + B))?1, the bouns are as claime. Finally, P assume that fx i g is a basis for. Then fx i g an fy i g are biorthonormal, so T x j = ht?1 T x j ; y i i w i = w j. Therefore fw i g is a basis since T is invertible. Conversely, if fw i g is a basis then T?1 maps it onto fx i g. In the terminology of Kato [K, p. 181], the hypotheses P in Theorem 2 are that the operator K: D(K)! ene P by Kfc i g = c i (x i? w i ) is relatively boune with respect to the operator fc i g 7! c i x i. It is natural to call K the \perturbation operator," since, as we have seen, conitions on K imply that \fw i g inherits ecomposition properties from fx i g." We point out some consequences of Theorem 2. First, specic choices of an give conitions in the style of classic results on basis perturbation. Corollary 3. Let (fy i g; fx i g) be an atomic ecomposition of with respect to with bouns A, B. Assume that, satisfy: (a) has an absolutely continuous norm, (b) is a soli Banach space of scalar-value sequences, an (c) the action of fc i g 2 on fb ig 2 is given by hfb i g; fc i gi = P b i c i.

6 6 ole christensen an christopher heil If fw i g is such that R = kfkx i? w i k gk < 1 B ; then there exists a family fz i g such that (fz i g; fw i g) is an atomic ecomposition of with respect to with bouns A (1 + R B)?1, B (1? R B)?1. Proof. The hypotheses imply that i c i (x i? w i ) R kfc i gk for any nite sequence fc i g 2. Therefore we can apply Theorem 2 with = an = R. A rawback of Corollary 3 is that it generally oes not apply to the problem of perturbing the mother wavelet of a coherent state atomic ecomposition. These are the most important practical incarnations of atomic ecompositions. A coherent state atomic ecomposition has x i = (g i )x, where is a representation of a group G on such that each (g) is a bijective isometry of onto itself, fg i g is a iscrete set in G, an x 2 is the generator or, by an abuse of terminology, the mother wavelet. (See [HW] for examples of typical groups an representations). When the mother wavelet x is perturbe, say to w, we have k(g i )x? (g i )wk kx? wk. Hence kfk(g i )x? (g i )wk gk will typically be innite. On the other han, the hypotheses of Theorem 2 can still be applicable (we iscuss this further in the Hilbert space setting following Corollary 6). Feichtinger an Grochenig [FG] have prove some perturbation results for coherent state atomic ecompositions. The novelty of Theorem 2 is its general formulation an proof. We say that a sequence fx i g is a Bessel sequence for with respect to if there exists a constant D such that kfhx i ; yigk D kyk for all y 2 : The constant D is the Bessel boun. The following aitional consequence of Theorem 2 is motivate by a useful result about Riesz bases in Hilbert spaces [Hi]. Corollary 4. Let (fy i g; fx i g) be an atomic ecomposition of with respect to with bouns A, B, an such that fx i g is a Bessel sequence for with respect to with

7 perturbations of banach frames 7 Bessel boun D. Assume that, satisfy hypotheses (a), (b), an (c) of Corollary 3. Assume that there exists a family ft k g of boune operators on an scalars a ik so that x i? w i = k a ik T k x i for each i: If (a) a k = sup i ja ik j < 1 for each k, an (b) P a k kt k k < (BD)?1, then there exists a family fz i g such that (fz i g; fw i g) is an atomic ecomposition of with respect to with bouns A (1 + BD P a k kt k k)?1, B (1? BD P a k kt k k)?1. Proof. Given a nite sequence fc i g 2, we have i c i (x i? w i ) = i c i Fix any k. Then c i a ik x i = sup i k kyk =1 a ik T k x i i k c i a ik hx i ; yi kt k k = sup jhfc i g; fa ik hx i ; yigij kyk =1 i c i a ik x i : kfc i gk sup kfa ik hx i ; yigk kyk =1 where we have use the fact that i c i (x i? w i ) D a k kfc i gk ; is soli. Hence D k a k kt k k kfc i gk for every nite sequence fc i g 2. We can therefore apply Theorem 2 with = an = D P a k kt k k. 3. Duality for atomic ecompositions. If fx i g is a basis for with coecient functionals fy i g then fy i g is a basis for spanfy i g with coecient functions fx i g. We investigate the analogous question for atomic ecompositions.

8 8 ole christensen an christopher heil Theorem 5. Let (fy i g; fx i g) be an atomic ecomposition of with respect to. Assume, satisfy: (a) is soli, (b) is a Banach space of scalar-value sequences, (c) the action of fc i g 2 on fb ig 2 is given by hfb i g; fc i gi = P b i c i, an () has an absolutely continuous norm. If fx i g is a Bessel sequence for with respect to then (fx ig; fy i g) is an atomic ecomposition of with respect to. Proof. The hypotheses given imply that P c i y i converges in for every fc i g 2. In particular, since fx i g is a Bessel sequence, if y 2 is xe then fhx i ; yig 2, so P hxi ; yi y i converges in. Moreover, if x 2 then D x; hxi ; yi y i E Hence P hx i ; yi y i = y for each y 2. = D hx; yi i x i ; ye = hx; yi: It remains only to show that there is a constant C such that C kyk for all y 2. However, if y 2 then kfhx i ; yigk kyk = sup jhx; yij kxk =1 = sup kxk =1 hx; y i i hx i ; yi sup kfhx; y i igk kfhx i ; yigk kxk =1 B kfhx i ; yigk ; so the proof is complete. The Bessel sequence hypothesis is clearly necessary. For example, suppose (fy i g; fx i g) is an atomic ecomposition of with respect to an that fw j g is not a Bessel sequence for with respect to. Dene z j = for each j; then (fy i g [ fz j g; fx i g [ fw j g) is an atomic ecomposition of with respect to, although (fx i g [ fw j g; fy i g [ fz j g) is not an atomic ecomposition of with respect to.

9 perturbations of banach frames 9 4. Frame ecompositions in Hilbert spaces. In this section we consier the case = H, a separable Hilbert space, an = `2. For Hilbert frames, it is customary to use a enition of frame bouns slightly ierent from the one we gave for Banach frames in Denition 2. In particular, if fx i g is a Hilbert frame then the norm equivalence between kxk H an kfhx; x i igk`2 is usually written A kxk 2 H i jhx; x i ij 2 B kxk 2 H for all x 2 H; (2) with these A, B calle the frame bouns. For clarity, we will refer to A, B given by (2) as Hilbert frame bouns; they are the squares of the Banach frame bouns given in Denition 2. First we prove an important consequence of Theorem 1. Corollary 6. Let fx i g be a Hilbert frame with Hilbert frame bouns A, B. Let fw i g H. If there is an R < A such that i jhx; x i? w i ij 2 R kxk 2 H for all x 2 H; (3) then fw i g is a Hilbert frame with Hilbert frame bouns A (1? p R=A) 2, B (1 + p R=B) 2. Proof. Let f ~x i g be the ual frame of fx i g. If we ene Sfc i g = P c i ~x i, then (fx i g; S) is a Banach frame for H with respect to `2 with Banach frame bouns p A, p B. By stanar Hilbert space arguments [DS], i c i ~x i H 1 p A kfc i gk`2 for every sequence fc i g 2 `2. Therefore, we can apply Theorem 1 with = an = p R to obtain ( p A? p 1=2 R) kxk H jhx; w i ij 2 ( p B + p R) kxk H : i In most cases it is more icult to verify the lower frame conition than the upper one. Corollary 6 shows that \the icult problem reuces to the easier one in the case of perturbation": the family fw i g is a frame if the ierence fx i? w i g satises the upper conition with a suciently small boun. Note that this is a weaker hypothesis than the

10 1 ole christensen an christopher heil stanar basis-type assumption that P kx i?w i k 2 H < A. In particular, this latter hypothesis cannot be applie to the problem of perturbing the mother wavelet x of a coherent state frame f(g i )xg. However, Corollary 6 oes apply to this problem: it states that f(g i )wg is a frame if the set of coherent states f(g i )(x? w)g generate by x? w is a Bessel sequence with boun less than A. As note above, establishing that f(g i )(x? w)g is a Bessel sequence is usually not a icult matter. For example, Favier an Zalik [FZ] obtain such results explicitly for the case of Gabor frames (frames where is the Schroeinger representation of the Heisenberg group on L 2 (R)). For applications of Corollary 6 to other problems in irregular sampling an wavelet theory, we refer to [FZ] an [C3]. We have alreay remarke on the importance of the perturbation operator K. For Hilbert frames we are able to prove another result where K plays the main role. Theorem 7. Let fx i g be a Hilbert frame for H, an let fw i g H. If Kfc i g = P ci (w i? x i ) is compact as an operator from `2 into H, then fw i g is a Hilbert frame for spanfw i g. Proof. Dene T : `2! H by T fc i g = P c i x i. Since fx i g is a frame, we know that T is boune. In fact, kt k 2 B, the upper Hilbert frame boun for fx i g. Hence V = T + K is a boune operator from `2 into H. If x 2 H then we compute jhx; wi ij 2 = kv xk 2 H kt + Kk 2 kxk 2 H B This establishes that fw i g satises an upper frame boun. 1 + kkk p kxk 2 H : B The hypothesis that K is compact will give us the existence of the lower frame boun, but it will not give a concrete value. By [C1, Theorem 2.1], to show the existence of the lower frame boun for fw i g, it suces to show that the \frame operator" V V for fw i g is surjective. Now, V V = S + T K + KT + KK ; where S = T T is the frame operator for fx i g. The operator (T K + KT + KK ) S?1 is compact, so the operator (T K +KT +KK ) S?1 +I has close range [R, Theorem 4.23]. Composing this with S, we see that V V also has close range.

11 perturbations of banach frames 11 Now consier V V as an operator on the close subspace spanfw i g. Here V V is injective: if x 2 spanfw i g an V V x = then P jhx; w i ij 2 = hv V x; xi =, whence x =. Since V V has a close range we therefore have Range(V V ) = (N(V V ))? = spanfw i g. Thus V V is surjective, as esire, an hence fw i g is a frame for spanfw i g. In particular, fw i g is a frame for spanfw i g if P kx i? w i k 2 H < 1. By Corollary 6 we know that if P kx i? w i k 2 H < A (the lower Hilbert frame boun for fx ig) then fw i g is a frame for H, an therefore spanfw i g = H. However, if we have merely the equality P kxi? w i k 2 H = A, it may happen that spanfw ig 6= H. For example, let fx i g be an orthonormal basis for H, an set w 1 =, w i = x i for i > 1. Also, note that the conition (3) in Corollary 6 is precisely the statement that kkk < p A. If kkk p A then fw i g nee not be a frame for spanfw i g. For example, if fx i g is an orthonormal basis for H an we set w i = x i + x i+1, then kkk = A = 1 but fw i g is not a frame for spanfw i g = H. Our nal result establishes the relation between Hilbert frames an atomic ecompositions in Hilbert spaces. Note that if fy i g is a Hilbert frame for H then (fy i g; f~y i g) is an atomic ecomposition of H with respect to `2, where f~y i g is the ual frame of fy i g. The converse requires aitional hypotheses. Theorem 8. Let (fy i g; fx i g) be an atomic ecomposition of H with respect to `2. Then the following statements hol. for H. (a) fy i g is a Hilbert frame for H. (b) If fx i g is a Bessel sequence for H with respect to `2 then it is a Hilbert frame (c) Assume fx i g is a Bessel sequence for H with respect to `2. Dene U; V : H! `2 by Ux = fhx; x i ig an V x = fhx; y i ig. Then fx i g is the ual frame of fy i g if an only if Range(U) = Range(V ). Proof. Statement (a) follows immeiately from the enition. For (b), the lower frame boun follows from Theorem 5, or irectly from the computation kxk 4 H = i hx; y i i hx i ; xi 2 i jhx; y i ij 2 i jhx i ; xij 2 B 2 kxk 2 H i jhx i ; xij 2 : Finally, for (c), note that the reconstruction formula (1) implies U V = V U = I.

12 12 ole christensen an christopher heil Let E = Range(U). Since U is injective an UV U = U, we have (UV )j E E = Range(V ), this implies UV V = V. Therefore, given x 2 H, = Ij E. If fhx; y i ig = V x = UV V x = fhv V x; x i ig = fhx; V V x i ig: In particular, we must have y i = V V x i, whence x i = (V V )?1 y i an fx i g is the ual frame of fy i g. Conversely, if fx i g is the ual frame of fy i g then V x = UV V x, so Range(V ) = Range(U) since V V is invertible. It nee not be the case that fx i g is the ual frame of fy i g even if (fy i g; fx i g) is an atomic ecomposition of H with respect to `2 an fx i g is a Bessel sequence. For example, let fy i g an fz j g be two frames for H. Dene w j = for each j. Then (fy i g [ fz j g; f~y i g [ fw j g) is an atomic ecomposition of H with respect to `2, but the ual frame of fy i g[fz j g is f~y i g [ f~z j g. We close with a note about convergence. The hypotheses on, use in most of the results were neee to ensure that series such as P c i y i converge unconitionally for every fc i g in the appropriate sequence space. In the Hilbert setting, we know that if fy i g is a Hilbert frame then P c i y i converges unconitionally in H for every fc i g 2 `2. In fact, this is true if fy i g is merely a Bessel sequence for H with respect to `2. Moreover, if fy i g is an arbitrary sequence in H an P c i y i converges unconitionally, then Orlicz' Theorem implies P jc i j 2 ky i k 2 H = P kc i y i k 2 H < 1. Therefore, if fy ig is norm-boune below (meaning inf ky i k H > ), then P jc i j 2 < 1. In particular, if fy i g is a Bessel sequence for H with respect to `2 an fy i g is normboune below, then fc i g 2 `2 () c i y i converges unconitionally in H: It woul be useful to similarly characterize unconitional convergence in the Banach space setting. Acknowlegments. We thank Fre Anrew, John Beneetto, Hans Feichtinger, Karlheinz Grochenig, Henrik Stetkr an Davi Walnut for valuable iscussions an insights.

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