COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES

Transcription

1 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK Abstract. Let e i ) be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces. Contents 1. Introduction 1 2. Frames in Hilbert spaces and Banach spaces 5 3. Three Examples Quantization with Z-bounded coefficients Quantization and Cotype Concluding remarks and open problems 35 References Introduction Hilbert space frames provide a crucial theoretical underpinning for compression, storage and transmission of signals because they provide robust and stable representation of vectors. They also have applications in mathematics and engineering in a wide variety of areas including Key words and phrases. Coefficient Quantization; Banach spaces; Frames Mathematics Subject Classification: Primary 46B20, Secondary 41A65. The research of the first, second, third and fourth author was supported by the NSF. The first, second, third, and fifth author was supported by the Linear Analysis Workshop at Texas A&M University in All authors were supported by the Banff International Research Station. 1

2 2P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK sampling theory [AG], operator theory [HL], harmonic analysis, nonlinear sparse approximation [DE], pseudo-differential operators [GH], and quantum computing [EF]. In many situations it is useful to think of a signal as being a vector x of a Hilbert space and being represented as a finite or infinite) sequence < x i, x >), where x i ) is a frame, i.e. a sequence in H which satisfies for some 0 < a b, 1) a x 2 < x i, x > 2 b x 2, whenever x H. Since the sequence x i ) does not have to be and usually is not) a basis for H, the representation of the element x as the sequence < x i, x >) includes some redundancy, which, for example, can be used to correct errors in transmissions [GKK]. Using a Hilbert space as the underlying space has, inter alia, the advantage of an easy reconstruction formula. Nevertheless, there are circumstances which make it necessary to leave the confines of a Hilbert space, and generalize frames to the category of Banach spaces. One such instance occurs when we wish to replace the frame coefficients by quantized coefficients, i.e. by integer multiples of a given δ > 0. An example of such a situation is described by Daubechies and De- Vore in [DD]: Let f L 2, ) be a band-limited function, to wit, the support of the Fourier Transform ˆf is contained in [ Ω, Ω] for some Ω > 0. For simplicity we assume that Ω = π. Now we can think of ˆf as an element of L 2 [ π, π], write ˆf on [ π, π] as a series in e inx, n Z, and apply the inversion formula for the Fourier transform. This leads to the sampling formula fx) = n Z fn) sinxπ nπ), x R. xπ nπ This series converges badly. In particular it is not absolutely convergent in general. Therefore we consider some λ > 1 and think of the space L 2 [ π, π] as being embedded in the natural way) into L 2 [ λπ, λπ]. The family of functions e inx/λ ) n Z forms an orthogonal basis for L 2 [ λπ, λπ], and it can be viewed as a frame for the smaller space L 2 [ π, π] see section 2). We write ˆfξ) = 2π ˆρξ) ˆfξ), where ˆρ : R [0, 1/ 2π] is C, ˆρ [ π,π] 1/ 2π, and ˆρ,λπ] [λπ, ) 0. Now we can express ˆf on [ λπ, λπ] as a series in e inx/λ ) and apply

3 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 3 the inverse transform once again. This leads to the expansion fx) = 1 n f ρ x λ λ) n ), x R, λ n Z which not only converges faster, but is also absolutely unconditionally convergent, since ˆρ is C, and, thus, ρ and all its derivatives are in L 1 R). Now assume that f L 1 note that bandlimited functions are bounded in L ). It was shown in [DD] that the Σ -quantization algorithm can be used to find a sequence q n ) n Z { 1, 1} for which fx) 1 q n ρ x n ) 1 λ λ λ ρ L1, for x R. n Z This means that our approximation does not hold in L 2 and it need not for the applications at hand) but it does hold in the Banach space L in fact in CR)). We consider therefore a signal to be an arbitrary vector x in a Banach space X and ask if there is a dictionary e i ), e.g. some sequence e i ) whose span is dense in X, so that x can be approximated, up to some ε > 0, by a linear combination of the e i s using only coefficients from a discrete alphabet, i.e. the integer multiples of some given δ. The case that e i ) is a non-redundant system, for example a basis, or, more generally, a total fundamental minimal system, was treated in [DOSZ]. It was shown there, for example, that if e i ) is a semi-normalized fundamental and total minimal system which has the property that for some ε, δ > 0 every vector of the form x = i E a ie i, with E N finite, can be ε-approximated by a vector x = i E δk ie i, with k i ) Z, then e i ) must have a subsequence which is either equivalent to unit-vector basis of c 0, or to the summing basis for c 0. Conversely, every separable Banach space X containing c 0 admits such a total fundamental minimal system. In this work we will concentrate on redundant dictionaries. Our model for redundant dictionaries will be frames in Banach spaces. In section 2 we shall recall their definition and make some elementary observations. Before we tackle the problem of coefficient quantization with respect to frames, we first have to ask ourselves what exactly we mean by a meaningful coefficient quantization. In section 3 we recall

4 4P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK the notion Net Quantization Property NQP) as introduced for fundamental systems in [DOSZ]. We shall then present several examples of systems which formally satisfy NQP, but on the other hand clearly do not accomplish the goals of quantization, namely data compression and easy reconstruction. These examples will lead us to a notion of quantization which is more restrictive, and more meaningful, in the case of redundant systems. In section 4 we ask under which circumstances one can approximate a vector in a Banach space X by a vector with quantized coefficients which are bounded in some associated sequence space Z with a basis z i ) see Definition 4.1). If Z has non trivial lower estimates this is only possible if one reconciles with the fact that the length of the frame increases exponentially with the dimension of the underlying space. We shall show this type of quantization cannot happen if z i ) satisfies nontrivial lower and upper estimates. The proof of these facts utilizes volume arguments and must therefore be formulated first in the finite-dimensional case. An infinite-dimensional argument proves directly that the associated space Z with a semi-normalized basis z i ) cannot be reflexive. In particular, there is no semi-normalized frame x i ) for an infinite-dimensional Hilbert space so that for some choice of 0 < ε, δ < 1 and C 1, every x H, x = 1, can be ε-approximated by a vector x = δk i x i, with k i ) Z and δ 2 ki 2 C. In section 5 we consider conditions under which an n-dimensional space admits, for given ε, δ > 0 and C 1, a finite frame x i ) N, so that every element in the zonotope { N a ix i : a i 1} can be ε- approximated by some element from { N δk ix i : k i Z, k i C/δ}. Using results from convex geometry we shall show that this is only possible for spaces X with trivial cotype. Among others, we provide an answer to a question raised in [DOSZ] and prove that l 1 does not have a semi-normalized basis with the NQP. In the final section we will state some open problems. All Banach spaces are considered to be spaces over the real field R. S X and B X, denote the unit sphere and the unit ball of a Banach space X, respectively. For a set S we denote by c 00 S), or simply c 00, if S = N, the set of all families x = ξ s ) s S with finite support

5 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 5 suppx) = {s S : ξ s 0}. The unit vector basis of c 00, as well as the unit vector basis of l p, 1 p <, and c 0 is denoted by e i ). A Schauder basis, or simply a basis, of a Banach space X is a sequence x n ), which has the property that every x can be uniquely written as a norm converging series x = a i x i. It follows then from the Uniform Boundedness Principle that the coordinate functionals x n), x n : X R, ai x i a n are bounded cf.[fhhmpz]) and the projections P n, with P n : X X, x = n a i x i a i x i, for n N are continuous and uniformly bounded in the operator norm. We call C = sup n N P n the basis constant of x i ) and K = sup 0 m n P n P m P 0 0) the projection constant of x i ). Note that C K 2C. We call x n ) monotone if C = 1 and bimonotone if also K = 1. A basis x n ) is called unconditional if for any x X the unique representation x = a n x n converges unconditionally. This is equivalent cf. [FHHMPZ]) to the property that for all a i ) c 00 { } K u = sup ±ai x i : ai x i = 1 <. If X is a finite dimensional space we can represent it isometrically as R n, ) where is a norm function on R n. With this representation we consider the Lebesgue measure of a measurable set A R n and denote it by VolA). Of course VolA) depends on the representation of X. Nevertheless, if we only consider certain ratios of volumes this is not the case. Therefore, the quotient VolA)/VolB) is well defined even in abstract finite dimensional spaces without any specific representation. 2. Frames in Hilbert spaces and Banach spaces In this section we give a short review of the concept of frames in Banach spaces, and make some preparatory observations. Let us start with the well known notion of Hilbert space frames. Definition 2.1. Let H be a finite or infinite dimensional) Hilbert space. A sequence x j ) j J in H, J = N or J = {1, 2,..., N}, for some

6 6P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK N N, is called a frame of H or Hilbert frame for H if there are 0 < a b < so that 2) a x 2 j N x, x j 2 b x 2 for all x H. For a frame x j ) j J of H we consider the operator Θ : H l 2 J), x x, x j ) j J, its adjoint Θ : l 2 J) H, ξ j ) j J j J ξ j x j and their product Since a x 2 j N I = Θ Θ : H H, x j J x, x j x j. x, x j 2 = x, x, x j x j = x, Ix) b x 2, j N I is a positive and invertible operator with aid H I bid H and thus, x = I 1 Ix) = j N x, x j I 1 x j ), or x = I I 1 x) = j N I 1 x), x j x j = j N x, I 1 x j ) x j. For an introduction to the theory of Hilbert space frames we refer the reader to [Ca1] and [Ch]. We follow [HL] and [CHL] for the generalization of frames to Banach spaces. Definition 2.2. Schauder Frame) Let X be a finite or infinite dimensional) separable Banach space. A sequence x j, f j ) j J, with x j ) j J X, f j ) j J X, and J = N or J = {1, 2... N}, for some N N, is called a Schauder) frame of X if for every x X 3) x = j J f j x)x j. In case that J = N, we mean that the series in 3) converges in norm, i.e. that x = lim n n j=1 f jx)x j. An unconditional frame of X is a frame x i, f i ) i N for X for which the convergence in 3) is unconditional.

7 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 7 We call a frame x i, f i ) bounded if sup i x i < and sup f i <, i and semi-normalized if x i ) and f i ) are semi-normalized, i.e. if 0 < inf i x i sup i x i < and 0 < inf i f i sup i f i <. In the following Remark we make some easy observations. Remark 2.3. Let x i, f i ) i N be a frame of X. 4) 5) w a) If inf i N x i > 0, then f i 0 as i. b) Using the Uniform Boundedness Principle we deduce that K = sup sup x B X m n n f i x)x i <. i=m This implies that if inf i N x i > 0 then f i ) is bounded and if inf i N f i > 0 then x i ) is bounded. We call K the projection constant of x i, f i ). The projection constant for finite frames is defined accordingly. c) For all f X and x X it follows that ) n fx) = f f i x)x i = f i x)fx i ) = lim fx i )f i )x), n n i=m and, thus, f = w fx i )f i. Moreover, for m n in N it follows that fx i )f i = sup x B X n fx i )f i x) f sup x B X i=m and fx i )f i = sup fx i )f i x) x B X i=m = sup x B X f i=m ) f i x)x i i=m n f i x)x i K f, i=m { sup z spanxi :i m), z K fz)=k f spanxi :i m) f {f1,f 2,...f m}.

8 8P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK d) If x i, f i ) is an unconditional frame it follows from the Uniform Boundedness Principle that K u = sup sup σi f i x)x i <. x B X σ i ) {±1} We call K u the unconditional constant of x i, f i ). The following Proposition is a slight variation of [CHL, Theorem 2.6]. Proposition 2.4. Let X be a separable Banach space and let x i ) i J X and f i ) i J X, with J = N or J = {1, 2,... N} for some N N. a) x i, f i ) i J is a Schauder frame of X if and only if there is a Banach space Z with a Schauder basis z i ) i J and corresponding coordinate functionals z i ), an isomorphic embedding T : X Z and a bounded linear surjective map S : Z X, so that S T = Id X i.e. X is isomorphic to a complemented subspace of Z), and Sz i ) = x i, for i J, and T z i ) = f i, for i J, with x i 0. Moreover S and T can be chosen so that S = 1 and T K, where K is the projection constant of x i, f i ), and z i ) can be chosen to be a bimonotone basis with z i = x i if i J, with x i 0. b) x i, f i ) i J is an unconditional frame of X if and only if there is a Banach space Z with an unconditional basis z i ) and corresponding coordinate functionals z i ), an isomorphic embedding T : X Z and a surjection S : Z X, so that S T = Id X, Sz i ) = x i, for i J, and T z i ) = f i for i J, with x i 0. Proof. a) part Assume that x i, f i ) i J is a frame of X and let K be the projection constant of x i, f i ) i J. We put J = {i J : x i 0}, denote the unit vector basis of c 00 J) by z i ) and define on c 00 J) the following norm Z. 6) Z X ) 1/2 a i z i = max a i x i + a 2 i for ai ) R. m n i J i J {m,m+1,...,n} i J\ J It follows easily that z i ) is a bimonotone basic sequence and, thus, a basis of the completion of c 00 J) with respect to Z, which we denote by Z.

9 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 9 The map S : Z X, aj z j a j x j, is linear and bounded with S = 1. Secondly, define T : X Z, x = i J f i x)x i = i J f i x)x i i J f i x)z i. Remark 2.3 b) yields for x X Z f i x)z i = sup m n i J = sup m n i J {m,m+1,...,n} i J {m,m+1,...,n} f i x)x i X f i x)x i X K x, and, thus, that T is linear and bounded with T K. Clearly it follows that S T = Id X, which implies that T is an isomorphic embedding and that S is a surjection. Finally, if zi ) are the coordinate functionals of z i ) we deduce for x X and i J { T zi )x) = zi f i x) if x i 0 T x)) = 0 if x i = 0, which finishes the proof of. In order to show the converse in a), assume that Z is a space with a basis z i ) i J and that S : Z X is a bounded linear surjection, and T : X Z an isomorphic embedding, with S T = Id X. Put x i = Sz i ) and f i = T zi ), for i J. Then for x X, ) x = S T x) = S z i T x))z i = T zi )x)sz i ) = f i x)x i, which implies that x i, f i ) i J is a frame of X. For the proof of b) we replace 6) by Z X ) 1/2. 7) a i z i = max σ i a i x i + a 2 i i J σ i ) {±1} i J i J\ J and note that arguments similar to those in the proof of a) show our claim b). Definition 2.5. Let x i, f i ) be a frame of a Banach space X and let Z be a space with a basis z i ) and corresponding coordinate functionals

10 10 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK zi ). We call Z, z i ) ) an associated space to x i, f i ) or a sequence space associated to x i, f i ) and z i ) an associated basis, if S : Z X, ai z i a i x i and T : X Z, x = f i x)x i f i x)z i are bounded operators. We call S the associated reconstruction operator and T the associated decomposition operator or analysis operator. In this case, following [Gr] we call the triple x i ), f i ), Z ) an atomic decomposition of X. Remark 2.6. By Proposition 2.4 the property of Banach space X to admit a frame is equivalent to the property of being isomorphic to a complemented subspace of a space Z with basis. It was shown independently by Pe lczyński [Pe] and Johnson, Rosenthal and Zippin [JRZ] see also [Ca2][Theorem 3.13]) that the later property is equivalent to X having the Bounded Approximation Property, where X is said to have the Bounded Approximation Property if there is a λ 1, so that for every ε > 0 and every compact set K X there is a finite rank operator T : X X with T leλ so that T x) x ε whenever x K. Remark 2.7. Let x j ) j J be a Hilbert frame of a Hilbert space H and let Θ and I be defined as in the paragraph following Definition 2.1. We choose Z to be l 2 J), S = Θ and x i 0 T = Θ I 1 : H l 2, x j J I 1 x), x j e j = j J x, I 1 x j ) e j and observe that S T = Id H, and for j J it follows that Se j ) = Θ e j ) = x j, and T e j )x) = x, I 1 x j ), x H) and, thus T e j ) = I 1 x j ). Thus, if x i ) is a Hilbert frame, then x i ), I 1 x i )) is a Schauder frame for which Z = l 2 J) together with its unit vector basis is an associated space. Conversely, let x i, f i ) be a Schauder frame of a Hilbert space H and assume that Z = l 2 J) with its unit vector basis is an associated space. Denote by T : H l 2 J) and S : l 2 J) H the associated decomposition, respectively reconstruction operator. Then it follows

11 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 11 for all x H xi, x 2 = Se i ), x 2 = e i, S x) 2 = S x) 2. Thus, since S is an isomorphic embedding of H into l 2, it follows that x i ) is a Hilbert frame. In the following observation we show, that we can always expand a frame by a bounded linear operator. Proposition 2.8. Let x i, f i ) be a frame of a Banach space X and let Z be a space with a basis z i ) which is associated to x i, f i ). Furthermore assume that Y is another space with a basis y i ) and let V : Y X be linear and bounded. We put Z = Z Y and define z i ) Z, x i ) X and f i ) X by { { z i/2, λ i/2 y i/2 ) x i/2 + λ i/2 V y i/2 ) z i = x i = z i+1)/2, λ i+1)/2 y i+1)/2 ) x i+1)/2 λ i+1)/2 V y i+1)/2 ) if i even if i odd, f i = { 1 f 2 i/2 if i even 1 2 i+1)/2 if i odd, where λ i = z i / y i, for i N. Then x i, f i ) is a frame of X, z i ) is a basis for Z and Z, z i )) is an associated space for x i, f i ). Proof. Let T : X Z and S : Z X be the associated decomposition and reconstruction operator, respectively. Note that the operators S : Z Y X, z, y) Sz) + V y) and T : X Z Y, x T x), 0) are bounded and linear and that S T = Id X and S z i ) = x i, for i N. It is easy to verify that z i ) is a basis of Z for which its coordinate functionals z i ) are given by denote the coordinate functionals of y i ) by yi )) { 1 z i 2 = z i/2, 1 λ i yi/2 ) if i even 1 2 z i+1)/2, 1 λ i yi+1)/2 ) if i odd. it follows for x X that { } T z i )x) = z i T fi/2 x)/2 if i even x)) = = f i+1)/2 x)/2 if i odd f i x), which yields T z i ) = f i, for i N. Thus, the claim follows from Proposition 2.4.

12 12 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK 3. Three Examples In [DOSZ] the following notion of quantization was introduced and studied for non redundant systems. Definition 3.1. Let x i ) i J be a fundamental system, with J = N or J = {1, 2,... N}, for some N N, and let ε > 0 and δ > 0 be given. We say that x i ) i J has the ε, δ)-net Quantization Property abbr. ε, δ)-nqp) if for any x = i E a ix i X, E J finite, there exists a sequence k i ) Z, suppk i ) = {i N : k i 0} is finite, such that 8) x k i δx i ε. We say that x i ) has the NQP if x i ) has the ε, δ)-nqp for some ε > 0 and δ > 0. When we ask whether or not in a certain representation of vectors the coefficients can be replaced by quantized coefficients, we are often interested in memorizing data as economically as possible, and reconstructing them with as little error as possible. With this in mind, we will exhibit in this section several examples, which show that it is not always meaningful to apply the notion of NQP word for word to redundant systems like frames. These examples will then also guide us to more appropriate quantization concepts for frames. The first example is a tight Hilbert frame f i ) in S l2 i.e. a = b) consisting of normalized vectors so that for every x l 2 there is a sequence k i ) Z so that x i N k if i < 1. The second example is a semi-normalized Hilbert frame f i ) in l 2 which has the property that for every x l 2 there is a sequence k i ) Z so that x i N k if i < 1 and k i ) has the additional property that max i N k i 1, if x B l2. The third example is a Schauder frame f i ) of l 2 which has the property that for every x l 2 there is a sequence k i ) Z, so that not only max i N k i x and x i N k if i < 1, but so that also the support of k i ), i.e. the set {i N : k i 0} is uniformly bounded. Example 3.2. Let 0 < ε i < 1/2. For i N define the following vectors f 2i 1 and f 2i in S l2. f 2i 1 = 1 ε 2 i e 2i 1 + ε i e 2i and f 2i = ε i e 2i ε 2 i e 2i.

13 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 13 Clearly f i ) is an orthonormal basis for l 2 and we let F = {e i : i N} {f i : i N}. Then F is a tight frame as is any finite union of orthonormal bases) and the sequence z i ) with z 2i 1 = e 2i 1 f 2i 1 = 1 1 ε 2 i )e 2i 1 ε i e 2i z 2i = e 2i f 2i = ε i e 2i ε 2 i )e 2i, for i N is an orthogonal basis and z 2i 1 = z 2i = Oε i ). Thus, if ε i ) converges fast enough to 0 it follows that for any x l 2 there is a family k i ) Z, with suppk i ) <, so that x k i z i = x k i e i f i ) < 1. Example 3.3. Our second example is a semi-normalized Hilbert frame x i ) in l 2 so that D = { k i x i : k i { 1, 0, 1} and {i : k i 0} is finite } is dense in B l2. Put c i ) = 1/2 i ) and partition the unit vector basis e i ) of l 2 into infinitely many subsequences of infinite length, say ei, j) : i, j N). Then our frame f k ) is defined to be the sequence: f 1 = c 1 e 1 + e1, 1), f 2 = e1, 1), f 3 = c 1 e 2 + e1, 2), f 4 = e1, 2), f 5 = c 2 e 1 + e2, 1), f 6 = e2, 1), f 7 =c 1 e 3 + e1, 3), f 8 =e1, 3), f 9 =c 2 e 2 + e2, 2), f 10 =e2, 2), f 11 =c 3 e 1 + e3, 1), f 13 =e3, 1), f 14 = c 1 e 4 + e1, 4), f 15 = e1, 4),..., f 20 = c 4 e 1 + e4, 1), f 21 = e4, 1),. Note that the set of vectors x B l2 of the form x = εi, j)c i e j = εi, j) c i e j + ei, j) ) εi, j)ei, j) i,j N i,j N where εi, j)) { 1, 0, 1} so that the set {i, j N : εi, j) 0} is finite, are dense in B l2. This implies that every x B l2 is the limit of vectors x n ) with x n ) { εi f i : ε i ) { 1, 0, 1}, {i N : ε i 0} is finite }.

14 14 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK The sequence f n ) is a frame. Indeed for any x = x i e i l 2 we have fi, x 2 = f i, x 2 = x 2 i + c i x j + ei, j), x ) 2 i, x i even f 2 + i odd i,j N { x 2 x 2 i + i,j N 2c2 i x 2 j + 2 ei, j), x c 2 i ) x 2. Example 3.4. We construct a Schauder frame x i, f i ) of l 2 so that x n ) is dense in B l2. Let z n ) be dense in B l2 and choose for each n N x 2n 1 = z n + e n, x 2n = z n, f 2n 1 = e n and f 2n = e n. Clearly, for every x l 2 x = e i, x e i = f 2n 1, x x 2n 1 + f 2n, x x 2n the above sum is conditionally converging). It follows that x n ), f n ) ) is a Schauder frame of l 2. It is clear that x n ) is not a Hilbert frame. Remark 3.5. All three examples satisfy the conditions NQP) if we extend this notion word for word to frames. Nevertheless these examples do not satisfy our understanding of what quantization of coefficients should mean. In Example 3.2 every x l 2 can be approximated by an expansion with respect to a frame using only integer coefficients, but these coefficients might get arbitrarily large for elements in B l2. This means that we would need an infinite alphabet to approximate vectors which are in B l2. Therefore it is not enough as in the non redundant case) to assume that our frame is semi-normalized. In the Examples 3.3 and 3.4 we achieve the approximation of any vector in l 2 by a quantized expansion whose coefficients are bounded by a fixed multiple of the norm of the vector, but in order to approximate even the vectors of a given finite dimensional subspace for example the space generated by two elements of the unit vector basis elements of l 2 ) we need an infinite dictionary.

15 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES Quantization with Z-bounded coefficients One way to avoid examples like the ones mentioned in Section 3 is to impose boundedness conditions on the quantized coefficients within an associated space Z. Definition 4.1. Assume x i, f i ) i J, J = N or J = {1, 2... N}, for some N N, is a frame of a Banach space X. Let Z be a space with basis z i ) which is associated to x i, f i ) i J. Let ε, δ > 0, C 1. We say that x i, f i ) satisfies the ε, δ, C)-Net Quantization Property with respect to Z, z i )) or ε, δ, C)-Z-NQP, if for all x X there exists a sequence k i ) i J Z with finite support so that Z 9) ki δz i C x and x X δk i x i ε. We say that x i, f i ) satisfies the NQP with respect to Z, z i )) if it satisfies the ε, δ, C)-Z-NQP for some choice of ε, δ > 0 and C 1. It is easy to see that the property ε, δ, C)-NQP with respect to some associated space is homogenous in ε, δ), meaning that a frame x i, f i ) is ε, δ, C)-NQP if and only if for some λ > 0 or for all λ) x i, f i ) satisfies the λε, λδ, C)-NQP. The following result, analogous to [DOSZ, Theorem 2.4], shows that it is enough to verify that one can quantize the coefficients of elments x which are in B X to deduce the NQP. Proposition 4.2. Assume that x i ) and z i ) are some sequences in Banach spaces X and Z, respectively, and assume that there are C 0 <, δ 0 > 0 and 0 < q 0 < 1, so that for all x B X there is a sequence k i ) Z, k i ) c 00 with 10) δ 0 k i z i C0 and x δ 0 k i x i q0. Then there are δ 1 > 0, and C 1 < only depending on δ 0, q 0 and C 0 so that for all x X there is a sequence k i ) Z, k i ) c 00, with 11) δ1 k i z i C1 x and x δ 1 k i x i 1. Proof. Choose n 1 N and q 1 so that n ) q 0 = q 1 < 1. n 1 and put δ 1 = δ 0 /n 1.

16 16 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK We first claim that for any 0 < δ δ 1 and any x B X there is a sequence k i ) Z N c 00 so that 13) ki δz i 2C0 and x δk i x i q1. Indeed, let δ δ 1 and x B X and choose n n 1 in N so that δ 0 < δ δ 0 n+1 n and k i ) Z so that δ 0 ki δ 0 z i C0 and δn + 1) x k i δ 0 x i q0 and, thus, since n n 1, δn + 1) ki n + 1)δz i = ki δ 0 z i n + 1 δ 0 n C 0 2C 0 and x δ 0 k i δn+1)x i n + 1)δ x δn + 1) δ k i δ 0 x i q 0 n+1) q 1. δ 0 δ 0 By induction on n N we show that for any δ q1 n 1 δ 1 and any x B X there is a k i ) Z, k i ) c 00, so that n 1 14) ki δz i 2C0 q1 i and x k i δx i q n 1. i=0 For n = 1 this is just 13). Assume our claim to be true for n and let δ δ 1 q1 n and x B X. By our induction hypothesis, we can find k i ) Z, k i ) c 00, so that n 1 ki δz i 2C0 q1 i and x k i δx i q n 1, i=0 Since q1 n x ki δx i ) B X and since δq n claim and choose k i ) Z N c 00 so that ki δq1 n z i 2C0 and q x n ) k i δx i and, thus 1 ki + k n 1 i )δz i 2C0 q1 i + 2C 0 q1 n, i=0 and x δki x i δ k i x i q n+1 1, 1 δ 1, we can use our first δq1 n k i x i q1, which finishes the induction step. Now define C 1 = 2C 0 n=0 qn 1 and let x X be arbitrary.

17 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 17 If x 1 this is the only case left to consider) we choose n N with q1 n < 1 x qn 1 1 and, by 14) we can choose k i ) Z N c 00 so that δ 1 ki x z n 1 i 2C 0 q1 i C 1 and x x k i δ 1 x x i q1 n, i=0 which yields ki δ 1 z i C1 q 1 x and x k i δ 1 x i q n 1 x 1. In the following result we consider a finite frame x i, f i ) N of a finite dimensional Banach space X, and exploit the fact that, if x i, f i ) N has the ε, δ, C)-NQP with respect to some space Z having a basis z i ) N, then the value ε n = VolB X )/VolεB X ) must be smaller then the cardinality of the set { F δ,c) x i ) = nj δx j : n j δz j C }. Proposition 4.3. Assume f, g : 0, ) 0, ) are strictly increasing functions so that 15) lim n fn) = and gn) ln n lim n n = 0, and C, B, R 1, and 0 δ, ε < 1. Assume that x i, f i ) N is a frame of a Banach space X with dimx) = n < and that Z is an N-dimensional space, N N, with basis z i ) N, which is associated to x i, f i ) N. Let T : X Z and S : Z X be the associated decomposition and reconstruction operator, denote by K Z the projection constant of z i ). Assume 16) F δ,c) x i ) is ε-dense in B X, and 17), f#a) n j z j whenever A {1, 2... N} and nj ) j A Z \ {0}. j A

18 18 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK Then 18) ln N n ln1/ε) f 1 C/δ) ln 4KZ C δ ) + 1. In addition there is an n 0 N depending on C, B, R, ε, δ, f and g) so that if, moreover, K Z B, S R and g#a) sup, 19) ±z j whenever A {1, 2... N}, j A then n n 0. Proof. First note that, if A {1, 2,.., N} and n j ) j A Z \ {0} with C j A n jδz Z j δf#a), then #A f 1 C/δ) and n j K Z C/δ for j A. Thus, 16) and the volume argument, mentioned before the statement of our proposition, yields ε n #F δ,c) x i ) N f 1 C/δ) ) 2KZ C δ ) f 1 C/δ) [ 2KZ C + 1 N δ which, after taking ln ) on both sides, implies 18). Now assume that also 19) is satisfied. Let e i, e i ) n be an Auerbach basis of X, i.e. e i = e i = 1 and e i e j ) = δ i,j). Such a basis always exists c.f [FHHMPZ, Theorem 5.6]). Choose 0 < η < so that ε1 + 1/η) < 1 and define for i = 1, 2... n { } A i = j {1, 2... N} : e i x j ) ε/ηk Z Cf 1 C/δ)n Then it follows for the right choice of σ j = ±1, j A i that g#a i ) ±z j j A i ±x j j A i ) σ j x j 1 S sup 1 S e i j A σ i #A i ε η S K Z Cf 1 C/δ)n + 1)] f 1 C/δ), and thus 20) #A i g#a i ) η S K ZCf 1 C/δ)n. ε

19 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 19 Put A = i n A i. If n j ) j N Z is such that N j=1 δn jx j F δ,c) x j ), then δn j x j j A c j A c,n j 0 n ) e i x j ) max δ n ε j #{j : n j 0} j N ηf 1 C/δ) ε η, where the second inequality follows from the definition of the A i s and the observation at the beginning of the proof, and the last inequality follows from the fact that 21) f#{j : n j 0}) nj z j C/δ. This implies together with 16) that the set { N } F δ,c) = n j δx j : n j ) Z, n j δz j C #A f 1 C/δ) δ j A is ε 1+ η) 1 -dense in BX. Hence, our usual argument comparing volumes and 21) yields 2KZ C ) ) f 1 C/δ) 2KZ C +1 j=1 #A f 1 C/δ) Taking ln ) on both sides and letting rl) = lnl)gl)/l for l N, we conclude by 20) and since ε1 + 1/η) < 1 that 1 ) n ln ε1 + 1/η) f 1 2KZ C )) C/δ) ln#a) + ln + 1 δ f 1 C/δ) ln n + max ln#a 4KZ C )) i) + ln i n δ = f 1 C/δ) ln n + max r#a #Ai ) 4KZ C )) i) + ln i n g#a i ) δ f 1 C/δ) ln n + nr#a i0 ) η S K ZCf 1 C/δ) 4KZ C )) + ln ε δ where i 0 n is chosen so that #A i0 is maximal. By our assumption on g we can find an l 0 N so that f 1 C/δ)rl) 1 2 ln 1 ), whenever l l 0. ε1 + 1/η) If A i0 l 0 then 1 ) n ln ε1 + 1/η) [ f 1 2KZ C C/δ) ln n + ln l 0 + ln δ δ ) f 1 C/δ) 1 +1 ε n 1 + 1/η), n )] + 1,

20 20 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK which implies that n is bounded by a number which only depends on ε, δ, C, f, g and K Z. If #A i0 > l 0, then it follows that n 2 ln 1 ) f 1 C/δ) ln n+ η S K ZCf 1 C/δ) 4KZ C )) +ln. ε1 + 1/η) ε δ which implies our claim also in that case. We shall formulate a corollary of Proposition 4.3 for the infinite dimensional situation. We need first to introduce some notation and make some observations. Let x i, f i ) i N be a frame of X. Furthermore assume that X has the π λ -property, which means that there is a sequence P = P n ) of finite rank projections, whose norms are uniformly bounded, and which approximate the identity, i.e. 22) x = lim n P n x), in norm for all x X. For example, if X has a basis e i ) we could choose for n N the projection onto the first n coordinates, i.e. n P n : X X, ai e i a i e i. It is easy to see that P n x i ), f i PnX)) is a frame of the space Xn = P n X). Moreover, condition 22) and a straightforward compactness argument shows that for any n N and any 1 < r < 1 there is an 2 M n = Mr, n) so that it follows that 23) N x f i, x P n x i ) 1 r) x, whenever x X n and N M n It follows that the operators Q n ), with Q n : X n X n, M n x Pnf i ), x P n x i ), are uniformly bounded Q n 2, for n N), invertible and their inverses are uniformly bounded Q 1 n 1, for n N). For x X r n we write x = Q 1 n Q n x) = M n Pnf i ), x Q 1 n P n )x i )

21 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 21 and deduce therefore that y n) i, g n) i ) Mn := Q 1 n P n x i ), P nf i ) ) M n is a finite frame of X n. Let Z be a space with basis z i ) which is associated to the frame x i, f i ). It follows easily that the operators S n ) and T n ) S n : Z n = [z i : i M n ] X n, M n T n : X n Z n, x = f i x)q 1 n M n M n z a i z i a i Q 1 n P n )x i ) M n P n )x i ) f i x)z i are uniformly bounded, and thus Z n is an associated space for the frame y n) i, g n) i ) i Mn while T n and S n are the associated decomposition and reconstruction operators, respectively. Finally assume that the frame x i, f i ) satisfies the ε, δ, C)-NQP with respect to Z. Again by compactness and using Proposition 4.2 we can choose M n = Mr, n) large enough so that it also satisfies 24) For all n N and all x X n there is a sequence k i ) Mn Z so that M n x M n k i δz i C x and δk i x i ε. After changing ε > 0 and δ proportionally, if necessary, and since r > 1 1 r, we can assume that q = + sup 2 r n P n ε < 1. For n in N and r x B Xn we can therefore choose k i ) Mn Z so that M n δk iz i C

22 22 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK and M n x δk i Q 1 n P n )x i ) Qn 1 Q n x) M n δk i P n x i ) Q 1 n Q n x) x + Q 1 n x M n Q 1 n Q n x) x + Q 1 n P n x 1 r r Thus, for every n N the frame P nf i ), Q 1 n + sup P n ε n r = q < 1. P n )x i ) ) M n satisfies condition 16) of Proposition 4.3 for ε = q). Therefore we deduce the following Corollary. Corollary 4.4. Let x i, f i ) i N be a frame of an infinite dimensional Banach space X for which there is a uniformly bounded sequence P n ) of finite rank projections which approximate the identity. Assume that x i, f i ) i N satisfies the ε, δ, C)-NQP with respect to a space Z with basis z i ) for some choice of ε > 0, δ > 0 and C so that q = 1 r + r sup n P n 2ε < 1 with 1 < r < 1. Let M r 2 n) be any sequence in N which satisfies 23) and 24). Finally assume that z i ) satisfies the following lower estimate { lim inf } n j z j : nj ) j A Z \ {0}, A N, #A = n =. n Then j A a) M n ) increases exponentially with the dimension of X n, i.e. there is a c > 1, so that M n c dimxn) eventually, lnn) { b) lim sup n n sup } ±z i : A N, #A = n, =. i A Let us simplify the conditions in Corollary 4.4 and observe that it implies the following. Corollary 4.5. Assume that X is an infinite dimensional Banach space with the π λ -property and that Z is a Banach space with a basis z i ) satisfying for some choice of 1 < q < p < lower l p and δk i P n x i ) M n δk i x i

23 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 23 upper l q estimates, which means that for some C 1 ) 1/p ai p ) 1/q. ai z i C ai q C Then no frame of X has the NQP with respect to Z, z i ) ). The following example shows how to construct a frame with respect to a space Z which contains l 1. Proposition 4.6. Let x i, f i ) i N be any frame of a Banach space X and let Z be a space with semi-normalized basis z i ), which is associated to x i, f i ). Then there is a frame x i, f i ) i N and a basis z i ) of Z = Z l 1 so that x i, f i ) i N has Z as an associated space and has the NQP with respect to Z. Moreover, x i, f i ) i N is semi-normalized if x i, f i ) i N has this property for example if x i ) is a normalized basis of X). Proof. Assume, without loss of generality that z i = 1 for i N. Choose a quotient map Q : l 1 X so that Qe i ) : i N) is a 1-2 net in B X and so that Qei ) ± x i > 1 for i N which is easy to 4 accomplish). Finally we apply Proposition 2.8 to Y = l 1 with its unit vector basis e i ) and V = Q, and observe that the frame x i, f i ) and basis z i ) of Z, as constructed there, has the property that for any x B X there is an i N so that x x 2i x 2i 1 = x Qe i ) and 1 2 z 2i z 2i 1 ) = 1 which implies by Proposition 4.2 that x i, f i ) i N has the NQP with respect to z i ). By construction of x i, f i ) in Proposition 2.8 it follows that f i ) is semi-normalized if f i ) has this property and since Qei ) ± x i > 1, 4 for i N, it follows that 1 4 x i sup x j + 1, j N which implies that x i, f i ) is semi-normalized if x i, f i ) has this property. Finally let us present an infinite dimensional argument implying that if Z is a reflexive space with basis it cannot be the associated space of a frame x i, f i ) i N, with x i = 1, for i N, which satisfies the NQP.

24 24 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK Proposition 4.7. Assume that Z is a reflexive space with a seminormalized basis z i ), and assume that x i, f i )) is a frame of an infinite dimensional Banach space X with associated space Z. Then x i ), f i )) cannot have the NQP with respect to Z. The following result follows from Proposition 4.7 as well as from Corollary 4.5. Corollary 4.8. A semi-normalized frame of an infinite dimensional Hilbert space H cannot have the NQP with respect to the associated Hilbert space l 2 N). Proof of Proposition 4.7. We assume w.l.o.g. that z i ) is bimonotone and let T : X Z and S : Z X be the associated decomposition and reconstruction operator, respectively. For C < and δ > 0 define { } B C,δ) = δk i z i Z : k i ) Z, δk i z i C. Assume that x i, f i ) has the NQP with respect to Z. Then we can choose δ > 0 small enough and C 1 large enough so that SB C,δ) ) is ε-dense in B X for some 0 < ε < 1. Since z i ) is semi-normalized and Z is reflexive, B C,δ) is weakly compact. Indeed, assume that for n N, y n = δk n) i z i B C,δ). After passing to subsequence we can assume that for all i N there is a k i N so that k n) i = k i whenever n i. Thus, by bimonotonicity, it follows that n δk iz i C, for all n N, and, thus, since z i ) is boundedly complete δk iz i Z and δk iz i C. Thus, δk iz i in B C,δ), and since k n) i converges point-wise to k i ) and z i ) is shrinking it is the weak limit of y n). The support of each element in B C,δ) is finite since z i ) is a semi-normalized basis, and thus B C,δ) is countable. Since SB C,δ) ) is ε-dense in X, it follows that the map ) E : X CB C,δ) ), with Ex ) δki z i = δk i S x )z i ), is an isomorphic embedding. Indeed for x B X there is an x B X so that x x) = 1 and a sequence k i ) Z c 00 so that x δk i x i ε,

25 and thus COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 25 Ex ) ) Ex ) ) ) δki z i = x δki x i = 1+x δki x i x 1 ε. But this would means that X is isomorphic to a subspace of the space of continuous functions on a countable compact space, and, thus, hereditarily c 0, which is impossible since X is a quotient of a reflexive space and thus also reflexive. 5. Quantization and Cotype In this section we consider a quantization concept for Schauder frames, which is independent of an associated space. Definition 5.1. Let x i, f i ) i J be a frame of a finite or infinite dimensional) Banach space X, J = N or J = {1, 2,... N}, for some N N, and let 0 < ε, 0 < δ 1 and 1 C <. We say that x i, f i ) i J satisfies the ε, δ, C)-Bounded Coefficient Net Quantization Property or ε, δ, C)-BCNQP if for all a i ) i J [ 1, 1] J c 00 J) there is a k i ) i J Z J c 00 J) so that a i x i i J i J δk i x i ε and max i J k i C δ. Remark 5.2. Let x i, f i ) i J be a frame of X and let 0 < ε, 0 < δ 1 and 1 C <. 25) a) Since for any a i ) i J c 00 J) and any i J we can write a i = m i δ + ã i with m i N, m i δ a i and ã i δ, for i J, x i, f i ) satisfies the ε, δ, C)-BCNQP implies that for all a i ) i J c 00 J) there is a k i ) i J Z J c 00 J) so that a i x i δk i x i ε and max k i max a i + C i J i J δ. i J i J a) immediately implies b) If x i, f i ) satisfies ε, δ, C)-BCNQP and 0 < λ 1 then x i, f i ) satisfies λε, λδ, 1 + λc)-bcnqp. c) If x i ) is a semi-normalized basis of X and f i ) are the coordinate functionals with respect to x i ) and x i ) satisfies the ε, δ)- NQP Definition 3.1), then x i, f i ) satisfies the ε, δ, C)-BCNQP

26 26 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK with C = 1 + ε sup i J f i. Indeed, for x X, x = a ix i, with a i 1, there is a sequence k i ) Z with finite support so that x δk i x i ε and δ max k i max f i x) + i f i x ) ) δk i x i 1 + ε sup f i. We will connect the property BCNQP with properties of the cotype of the Banach space. Definition 5.3. Let p 2. We say that a Banach space X has type p if there is a c < so that for all n N and all vectors x 1, x 2,... x n X n 2) 1/2 ave ±x i = 2 n n 2) 1/2 n ) 1/p. σ i x i c x i p σ i ) n {±1}n In that case the smallest such c will be denoted by T p X). Let q 2. We say that a Banach space X has cotype q if there is a c < so that for all n N and all x 1, x 2,... x n X: n ) 1/q x i q c ave n ±x i 2) 1/2 = c 2 n σ i ) n {±1}n i J n 2) 1/2. σ i x i The smallest of all these constants will be denoted by C q X). We say that X has only trivial type, or only trivial cotype if T P X) = for all p > 1, or C q X) =, for all q <. Basic properties of spaces with type and cotype can be found for example in [DJT] or [Pi2]. We are mainly interested in estimates of the volume ratio of the unit ball B X of a finite dimensional space X using C q X) and the connection between finite cotype and the lack of containing l n s uniformly. Assume X is an n-dimensional space which we identify with R n, ). Let E be the John ellipsoid of the unit ball B X of X, i.e. the ellipsoid contained in B X having maximal volume. It was show in [Jo] see also [Pi2, Chapter 3]) that E is unique. We call the ratio Vol 1/n B X )/Vol 1/n E) the volume ratio of B X. Combining [Ro, Theorem 6], which establishes an upper estimate for the volume ratio using T p X ), with a result of Maurey and Pisier [MP1, MP2] see also [DJT, Proposition 13.17]) estimating T p X ) and a result of Pisier [Pi1] see also [Pi2, Theorem 2.5]) estimating the K-convexity constant KX) of

27 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 27 X, we obtain the connection between the volume ratio of B X and the cotype constant of X. Theorem 5.4. There is a universal constant d so that for all finite dimensional Banach spaces X, with n = dimx) 2, and all 2 q <. ) 1/n VolBX ) 26) dc q X)n αq) ln n, VolE) where E X is the John ellipsoid of B X and αq) := q. We will also need a second upper estimate for the volume ratio due to Milman and Pisier [MiP]. Theorem 5.5. [MiP]see also [Pi2, Theorem 10.4]) There is a universal constant A so that for any finite dimensional Banach space X, ) 1/n VolBX ) 27) gc 2 X)) := AC 2 X) ln1 + C 2 X)) VolE) where E X is the John ellipsoid of B X. The next result describes the connection between the property of having a finite cotype for q < and the lack of of containing l n s uniformly. Theorem 5.6. [MP1, MP2] For N N there is a qn) 2, ) and a CN) < so that: 28) For any finite or infinite dimensional) Banach space X which does not contain a 2-isomorphic copy of l N we have that C qn) X) CN). Finally we will need the following result from [Os]. It is implicitly already contained in [GMP, pp.95 97], and it has probably been known for much longer. In order to state it we will need the following notation. Let m n N and let L R n be an m-dimensional subspace. Let Q n be the unit cube in R n. By a simple compactness argument there is a projection P : R n L for which VolP Q n )) is minimal. In that case we call the image P Q n ) a minimal-volume projection of Q n onto L.

28 28 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK Theorem 5.7. [Os, Theorem 1] Let L be a linear subspace of R n, and let M be the set of all minimal volume projections of Q n onto L. Then M contains a parallelepiped. We are now in the position to state and to prove the connection between cotype and BCNQP in the finite dimensional case. Theorem 5.8. There is a map n 0 : [1, ) 2 [0, ) so that for all finite dimensional Banach spaces X the following holds. If x i, f i ) N is a frame of X, with x i = 1, for i N, and which satisfies 1, δ, C)-BCNQP for some 0 < δ < 1 and C 1, then for all 2 q < 1 C ) N dimx) lndimx)) 2q ln ), whenever dimx) n C 0 δ, KC qx), δ where K is the projection constant of x i, f i ) N. Proof. Let Z be the space with a basis z i ) and let T : X Z and S : Z X be the associated decomposition and the reconstruction operator as constructed in the proof of Proposition 2.4 a). Since the x i s are normalized the z i s are also of norm 1. After a linear transformation we can assume that Z = R N and z i ) N is the unit vector basis of R N. Since z i ) is a bimonotone basis it follows that zi = 1, for i N. Hence B Z Q N, where Q N denotes the unit cube in R N. Define L = T X) and put n = dimx) = diml). Since S T = Id X it follows that P = T S is a projection from Z onto L and if we denote the John ellipsoid of T B X ) by E and we deduce that recall that by Proposition 2.4 a) T K) 29) E T B X ) = P T B X ) P K B Z ) P K Q N ). By Theorem 5.7 there is a minimal -volume projection M of Q N onto L which is a parallelepiped. Let B n denote the n dimensional Euclidean ball in R n. Since there is a universal constant c so that ) n c VolB n ), n and since 1 K E 1 T E L B Z L Q N M,

29 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 29 we deduce from the fact that B n is the John ellipsoid of the unit cube in R n [Jo] see also [Pi2, Chapter 3]), that ) 1/n ) 1/n ) 1/n VolP Q N )) K Vol E ) VolP Q N )) = Vol VolM) n 1 E) Vol 1 E) c K K The last inequality follows from applying a linear transformation A to L so that AM) is a the unit cube in L with respect to some orthonormal basis of L) and, thus A 1 E) is an ellipsoid whose volume cannot exceed K that of the Euclidean unit ball in L. Since T : X, ) L, T BX )), where T BX ) is the Minkowski functional for T B X ), is an isometry it follows from Theorem 5.4 that 30) Vol 1/n T B X )) dc q X)n αq) lnn)vol 1/n E) dckc q X)n 1 q lnn)vol 1/n P Q N )) the universal constant d was introduced in Theorem 5.4). Since the zonotope { N }) { N } P Q N ) = T S a i z i : a i 1 = a i T x i ) : a i 1, contains at most 1 + 2C δ )N points from the set D = { δn i T x i ) : n i ) Z, max δ n i C} and since from our assumption that x i f i ) N satisfies the 1, δ, C)-BCNQP it follows that we deduce that 1 + 2C δ and, thus, P Q N ) z D z + T B X ), ) N VolP Q N )) VolT B X )) n 1/q KdcC q X) lnn) N n lnn) q ln ) n lnlnn)) 1 + C ln ) n lndckc qx)) 1 + 2C ln ), 1 + 2C δ δ δ which easily implies our claim. In the next result we will show that, up to a constant factor, the result in Theorem 5.8 is sharp. We are using the simple fact that for any number 0 r 1 and any m N, r can be approximated by a finite sum of dyadic numbers, say r = m j=1 σ j2 j, σ j {0, 1}, for j = 1,... m, so that r r 2 m. ) n

30 30 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK Proposition 5.9. Let X be an n-dimensional space with an Auerbach basis e i, e i ) n. Let m N and let K be the projection constant of e i ) n. Then there is a frame x i,j,s), f i,j,s) : 1 i n, 1 j m, s = 0, 1) ordered lexicographically) so that 31) 1 2 x i,j,s) 2 and f i,j,s) = 1, whenever 1 i n, 1 j m and s=0, 1, 32) a i,j,s) :1 i n, 1 j m,s=0,1) [ 1, 1] k i,j,s) :i n,j m,s=0,1) { 3, 2,..., 3} n m 1 n m 1 2 m a i,j,s) x i,j,s) k i,j,s) x i,j,s) 1 + n 1 2 m j=1 s=0 j=1 s=0 i.e. xi,j,s), f i,j,s) : 1 i n, 1 j m, s=0,1) satisfies the 1 + n 2 m 1 2, 1, 3) -BCNQP ). m 33) The projection constant of x i,j,s), f i,j,s) : 1 i n, 1 j m, s=0,1) does not exceed 4K. Proof. For 1 i n and 1 j m define x i,j,0) = e 1, x i,j,1) = e j e 1 2 m i, f i,j,0) = e i and f i,j,1) = e i. Since for every x X n n m x = e i x)e i = e 2 j n m i x)e i 1 2 = f m i,j,1) x)x i,j,1) + f i,j,0) x)x i,j,0), j=1 x i,j,s), f i,j,s) : 1 i n, 1 j m, s = 0, 1) is a frame of X and it satisfies 31). In order to verify 32) let a i,j,s) : 1 i n, 1 j m, s = 0, 1) [ 1, 1] be given. For i = 1, 2,..., n it follows that m j=1 a 2 j i,j,1) 1 2 1, and, thus, we can choose m ki,j,1) : j m) {0, ±1} so that for each i n 34) m 2 j a i,j,1) 1 2 m k m i,j,1) j=1 j=1 2 j j=1 2 m 1 2 m 1 2. m Since the absolute value of M = n m j=1 ai, j, 1) + ai, j, 0) k i,j,1) is at most 3nm we can choose for 1 i n and 1 j m, k i,j,0) { 3, 2,..., 2, 3} so that a = M n m j=1 k i,j,0), has absolute

31 COEFFICIENT QUANTIZATION FOR FRAMES IN BANACH SPACES 31 value at most 1. We compute n m 1 a i,j,s) x i,j,s) j=1 s=0 n n m j=1 1 k i,j,s) x i,j,s) s=0 m 2 j n a i,j,1) e i 1 2 m j=1 n n m 2 j k i,j,1) e i 1 2 m j=1 m a i,j,1) + a i,j,0) k i,j,0) k i,j,1) j=1 m 2 j 2 m a i,j,1) k i,j,1) ) 1 + n 1 2 m 1 2 m j=1 which proves 32). To estimate the projection constant of x i,j,s), f i,j,s) : 1 i n, 1 j m, s=0, 1) we denote by lex the lexicographic order on {i, j, s) : i n, j m, s = 0, 1}, and let n n m 2 j n m x = a i e i = a i e 1 +a i e 1 +a i e i 1 2 m = f i,j,s) x)x i,j,s) j=1 and i 0, j 0, s 0 ) lex i 1, j 1, s 1 ). Then, if i 0 < i 1, f i,j,s) x)x i,j,s) i 0,j 0,s 0 ) lex i,j,s) lex i 1,j 1,s 1 ) = 2 1 {s0 =1} [a j 0 ] i0 e 1 + a i0 1 2 m + + i 1 1 i=i 0 +1 j=1 j 1 j=1 2 a i0 + + m j=j 0 +1 m 2 j a i e 1 + a i e 1 + a i j=1 s=0,1 2 j a i0 e 1 + a i0 e 1 + a i0 1 2 e m i m e i a i1 e 1 + a i1 e 1 + a i1 2 j 1 2 m e i 1 ) 1 {s1 =0}a i1 e 1 i 1 1 i=i 0 +1 a i e i + ai1 4K x. If i 0 = i 1 similar estimates give the to the same result for the remaining cases and 33) follows. Remark If we choose in Proposition 5.9 m = 2 log n and thus 2 m 1/n 2 we obtain a frame for X of approximate size 4n log 2 n) )

32 32 P. G. CASAZZA, S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND A. ZSÁK having the 3, 1, 3)-BCNQP. Thus as we mentioned earlier, up to a constant Theorem 5.8 is best possible. Remark In Theorem 5.8 we assumed for simplicity that the x i s of our frame are normalized. It is easy to see that the same proof works for a general frame, in that case n 0 depends also on a = min{ x i : i N, x i 0} and b = max{ x i : i N}. With a similar proof to that of Theorem 5.8 we derive an upper estimate for min i N x i, i N, assuming that x i, f i ) N is a frame of an n dimensional space X which satisfies the 1, δ, C)-BCNQP for some choice of δ > 0 and C < assuming that N is proportional to n. Theorem For any choice of δ 0, 1], and C, K, q, c 2 1 there is a value h = hδ, C, K, q, c 2 ) so that the following holds for all n N. If X is an n-dimensional space, N qn and x i, f i ) N is a frame of X with projection constant K which has the 1, δ, C)-BCNQP, then if C 2 X) c 2, min x i hδ, C, K, q, c 2). i N n Sketch of proof. Let x i, f i ) N be a frame of X, N qn, which has the 1, δ, C)-BCNQP and projection constant K. As in the proof of Theorem 5.8 we let Z be the associated space with basis z i ) which was constructed in Proposition 2.4, T : X Z the associated decomposition operator, and S the associated reconstruction operator. Let L = T X), and P = T S, and let us also assume that Z = R N and z i = e i for i N. Note that now z i = x i and z i = x i 1 and we can therefore follow the proof of Theorem 5.8 replacing Q N by the box Q N = N [ 1 x i, 1 ]. x i As in the proof of Theorem 5.8 it follows that 1 K T B X) P B Z ) P Q N ). For the John ellipsoid E of T B X ) it follows therefore that 1 K E M, where a M is a minimal volume projection of Q N which is also a parallelepiped in L, and as before we deduce that KVol 1/n P Q N )) Vol 1/n E) n/c. Instead of applying Theorem 5.4 we now use Theorem