Uncertainty principles for orthonormal sequences

Transcription

1 Uncertainty principles for orthonormal sequences Philippe Jaming Université d Orléans, Faculté des Sciences, MAPMO-Fédération Denis Poisson, BP 6759, F Orléans Cedex, France Alexander M. Powell Department of Mathematics, Vanderbilt University, Nashville, TN 3740, USA Abstract The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L (R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L (R) then the sequence is finite. In a related result, Shapiro also proved that if the elements of an orthonormal sequence and their Fourier transforms have uniformly bounded means and dispersions then the sequence is finite. This paper gives quantitative bounds on the size of the finite orthonormal sequences in Shapiro s uncertainty principles. The bounds are obtained by using prolate spheroïdal wave functions and combinatorial estimates on the number of elements in a spherical code. Extensions for Riesz bases and different measures of time-frequency concentration are also given. Key words: Uncertainty principle, spherical code, orthonormal basis, Hermite functions, prolate spheroïdal wave functions, Riesz basis. addresses: Philippe.Jaming@univ-orleans.fr (Philippe Jaming ), alexander.m.powell@vanderbilt.edu (Alexander M. Powell ). Partially supported by a European Commission grant on Harmonic Analysis and Related Problems IHP Network (Contract Number: HPRN-CT HARP), by the Balaton program EPSF, and by the Erwin Schrödinger Insitute. Partially supported by NSF DMS Grant and by an Erwin Schrödinger Institute Junior Research Fellowship Preprint submitted to Elsevier Science 30 June 006

2 Introduction The uncertainty principle in harmonic analysis is a class of theorems which state that a nontrivial function and its Fourier transform can not both be too sharply localized. For background on different appropriate notions of localization and an overview on the recent renewed interest in mathematical formulations of the uncertainty principle, see the survey []. This paper will adopt the broader view that the uncertainty principle can be seen not only as a statement about the time-frequency localization of a single function but also as a statement on the degradation of localization when one considers successive elements of an orthonormal basis. In particular, the results that we consider show that the elements of an orthonormal basis as well as their Fourier transforms can not be uniformly concentrated in the time-frequency plane. Hardy s Uncertainty Principle [4] may be viewed as an early theorem of this type. To set notation, define the Fourier transform of f L (R) by f(ξ) = f(t)e iπtξ dt, and then extend to L (R) in the usual way. Theorem. (Hardy s Uncertainty Principle) Let a, b, C, N > 0 be positive real numbers and let f L (R). Assume that for almost every x, ξ R, f(x) C( + x ) N e πa x and f(ξ) C( + ξ ) N e πb ξ. () The following hold: If ab > then f = 0. If ab = then f(x) = P (x)e πa x for a polynomial P of degree at most N. This theorem has been further generalized where the pointwise condition () is replaced by integral conditions in [3], and by distributional conditions in [9]. Also see [3] and [5]. One may interpret Hardy s theorem by saying that the set of functions which, along their Fourier transforms, is bounded by C( + x ) N e π x is finite dimensional, in the sense that its span is a finite dimensional subspace of L (R). In the case ab <, the class of functions satisfying the condition () has been fully described by B. Demange [9]. In particular, it is an infinite dimensional subset of L (R). Nevertheless, it can not contain an infinite orthonormal sequence. Indeed, this was first proved by Shapiro in []:

3 Theorem. (Shapiro s Umbrella Theorem) Let ϕ, ψ L (R). If {e k } L (R) is an orthonormal sequence of functions such that for all k and for almost all x, ξ R, then the sequence {e k } is finite. e k (x) ϕ(x) and ê k (ξ) ψ(ξ), Recent work of A. De Roton, B. Saffari, H.S. Shapiro, G. Tennenbaum, see [0], shows that the assumption ϕ, ψ L (R) can not be substantially weakened. Shapiro s elegant proof of Theorem. uses a compactness argument of Kolmogorov, see [3], but does not give a bound on the number of elements in the finite sequence. A second problem of a similar nature studied by Shapiro in [] is that of bounding the means and variances of orthonormal sequences. For f L (R) with f =, we define the following associated mean µ(f) = Mean( f ) = t f(t) dt, and the associated variance (f) = Var( f ) = t µ(f) f(t) dt. It will be convenient to work also with the dispersion (f) (f). In [], Shapiro posed the question of determining for which sequences of real numbers {a n } n=0, {b n } n=0, {c n } n=0, {d n } n=0 R there exists an orthonormal basis {e n } n=0 for L (R) such that for all n 0 µ(e n ) = a n, µ(ê n ) = b n, (e n ) = c n, (ê n ) = d n. Using Kolmogorov s compactness argument, he proved the following, []: Theorem.3 (Shapiro s Mean-Dispersion Principle) There does not exist an infinite orthonormal sequence {e n } n=0 L (R) such that all four of µ(e n ), µ(ê n ), (e n ), (ê n ) are uniformly bounded. An extension of this theorem in [9] shows that if {e n } n=0 is an orthonormal basis for L (R) then two dispersions and one mean (e n ), (ê n ), µ(e n ) can not all be uniformly bounded. Shapiro recently pointed out a nice alternate proof of this result using the Kolmogorov compactness theorem from []. The case for two means and one dispersion is different. In fact, it is possible to construct an orthonormal basis {e n } n=0 for L (R) such that the two means and one dispersion µ(e n ), µ(ê n ), (e n ) are uniformly bounded, see [9]. 3

4 Although our focus will be on Shapiro s theorems, let us also briefly refer the reader to some other work in the literature concerning uncertainty principles for bases. The classical Balian-Low theorem states that if a set of lattice coherent states forms an orthonormal basis for L (R) then the window function satisfies a strong version of the uncertainty principle, e.g., see [7,]. For an analogue concerning dyadic orthonormal wavelets, see []. Overview and main results The goal of this paper is to provide quantitative versions of Shapiro s Mean- Dispersion Principle and Umbrella Theorem, i.e., Theorems. and.3. Section addresses the Mean-Dispersion Theorem. The main results of this section are contained in Section.3 where we prove a sharp quantitative version of Shapiro s Mean-Dispersion Principle. This result is sharp, but the method of proof is not easily applicable to more general versions of the problem. Sections. and. respectively contain necessary background on Hermite functions and the Rayleigh-Ritz technique which is needed in the proofs. Section.4 proves a version of the mean-dispersion theorem for Riesz bases. Section 3 addresses the Umbrella Theorem and variants of the Mean-Dispersion Theorem. The main results of this section are contained in Section 3.4 where we prove a quantitative version of the Mean-Dispersion Principle for a generalized notion of dispersion, and in Section 3.5 where we prove a quantitative version of Shapiro s Umbrella Theorem. Explicit bounds on the size of possible orthonormal sequences are given in particular cases. Since the methods of Section are no longer easily applicable here, we adopt an approach based on geometric combinatorics. Our results use estimates on the size of spherical codes, and the theory of prolate spheroïdal wavefunctions. Section 3. contains background results on spherical codes, including the Delsarte, Goethals, Seidel bound. Section 3. proves some necessary results on projections of one set of orthonormal functions onto another set of orthonormal functions. Section 3.3 gives an overview of the prolate spheroïdal wavefunctions and makes a connection between projections of orthonormal functions and spherical codes. Section 3.6 concludes with extensions to Riesz bases. Growth of means and dispersions In this section, we use the classical Rayleigh-Ritz technique to give a quantitative version of Shapiro s Mean-Dispersion Theorem. We prove that, in this sense, the Hermite basis is the best concentrated orthonormal basis of L (R). 4

5 . The Hermite basis Results of this section can be found in []. The Hermite functions are defined by ( h k (t) = /4 ) k ( k d e πt e k! π dt) πt. It is well known that the Hermite functions are eigenfunctions of the Fourier transform, satisfy ĥk = i k h k, and form an orthonormal basis for L (R). Let us define the Hermite operator H for functions f in the Schwartz class by It is easy to show that Hf(t) = 4π d dt f(t) + t f(t). ( ) k + Hh k = h k, () π so that H may also be seen as the densely defined, positive, self-adjoint, unbounded operator on L (R) defined by Hf = k + π f, h k h k. From this, it immediately follows that, for each f in the domain of H Hf, f = k + π f, h k = = µ(f) f + (f) + µ( f) f + ( f). t f(t) dt + ξ f(ξ) dξ (3). The Rayleigh-Ritz Technique The Rayleigh-Ritz technique is a useful tool for estimating eigenvalues of operators, see [0, Theorem XIII.3, page 8]. Theorem. (The Rayleigh-Ritz Technique) Let H be a positive selfadjoint operator and define λ k (H) = sup ϕ 0,,ϕ k inf Hψ, ψ. ψ [ϕ 0,,ϕ k ], ψ,ψ D(H) where D(H) is the domain of H. Let V be a n + dimensional subspace, V D(H), and let P be the orthogonal projection onto V. Let H V = P HP and let H V denote the restriction of H V to V. Let µ 0 µ µ n be the 5

6 eigenvalues of H V Then λ k (H) µ k, k = 0,, n. The following corollary is a standard and useful application of the Rayleigh- Ritz technique. For example, [6, Chapter ] contains a version in the setting of Schrödinger operators. Corollary. Let H be a positive self-adjoint operator, and let ϕ 0,, ϕ n be an orthonormal set of functions. Then n n λ k (H) Hϕ k, ϕ k. (4) PROOF. If some ϕ k / D(H) then positivity of H implies that (4) trivially holds since the right hand side of the equation would be infinite. We may thus assume that ϕ 0,, ϕ n D(H). Define the n + dimensional subspace V = span {ϕ k } n and note that the operator H V is given by the matrix M = [ Hϕ j, ϕ k ] 0 j,k n. Let µ 0,, µ n be the eigenvalues of H V, i.e., of the matrix M. By Theorem., n n n λ k (H) µ k = Trace(M) = Hϕ k, ϕ k which completes the proof of the corollary..3 The Sharp Mean-Dispersion Principle Theorem.3 (Mean-Dispersion Principle) Let {e k } be any orthonormal sequence in L (R). Then for all n 0, n ( (e k ) + (ê k ) + µ(e k ) + µ(ê k ) ) (n + )(n + ). (5) 4π Moreover, if equality holds for all 0 n n 0, then there exists {c k } n 0 n=0 C such that c k = and e k = c k h k for each 0 k n 0. PROOF. Since H is positive and self-adjoint, one may use Corollary.. It follows from Corollary. that for each n 0 one has n k + π n He k, e k. (6) 6

7 From (3), note that since e k =, He k, e k = (e k ) + (ê k ) + µ(e k ) + µ(ê k ). This completes the proof of the first part. Assume equality holds in (5) for all n = 0,..., n 0, in other terms that, for n = 0,..., n 0, He n, e n = (e n ) + (ê n ) + µ(e n ) + µ(ê n ) = n + π. Let us first apply (3) for f = e 0 : k + π e 0, h k = He 0, e 0 = π = π e 0, h k since e 0 =. Thus, for k, one has e 0, h k = 0 and hence e 0 = c 0 h 0. Also e 0 = implies c 0 =. Next, assume that we have proved e k = c k h k for k = 0,..., n. Since e n is orthogonal to e k for k < n, one has e n, h k = 0. Applying (3) for f = e n we obtain that, k=n k + π e n, h k = k + π e n, h k = He n, e n n + = k=n π e n, h k. = n + π Thus e n, h k = 0 for k > n. It follows that e n = c n h n. Example.4 For all n 0, the Hermite functions satisfy µ(h n ) = µ(ĥn) = 0 and (h n ) = (ĥn) = n + 4π. For comparison, let us remark that Bourgain has constructed an orthonormal basis {b n } n= for L (R), see [4], which satisfies (b n ) +ε and π ( b n ) + ε. However, it is difficult to control the growth of µ(b π n), µ( b n ) in this construction. For other bases with more structure, see the related work in [] that constructs an orthonormal basis of lattice coherent states {g m,n } m,n Z for L (R) which is logarithmically close to having uniformly bounded dispersions. The means (µ(g m,n ), µ( g m,n )) for this basis lie on a translate of the lattice Z Z. It is interesting to note that if one takes n = 0 in Theorem.3 then this yields the usual form of Heisenberg s uncertainty principle (see [] for equivalences between uncertainty principles with sums and products). In fact, using (3), 7

8 Theorem.3 also implies a more general version of Heisenberg s uncertainty principle that is implicit in []. In particular, if f L (R) with f = is orthogonal to h 0,..., h n then (f) + ( f) + µ(f) + µ( f) n + π. For instance, if f is odd, then f is orthogonal to h 0, and µ(f) = µ( f) = 0. Using the usual scaling trick, we thus get the well known fact that the optimal constant in Heisenberg s inequality, e.g., see [], is given as follows (f) ( f) 4π f in general 3 4π f if f is odd. Corollary.5 Fix A > 0. If {e k } n L (R) is an orthonormal sequence and for k = 0,..., n, satisfies then n 8πA. µ(e k ), µ(ê k ), (e k ), (ê k ) A, PROOF. According to Theorem.3 n ( 4(n + )A (e k ) + (ê k ) + µ(e k ) + µ(ê k ) ) (n + )(n + ). 4π It follows that n + 6πA. This may also be stated as follows: Corollary.6 If {e k } L (R) is an orthonormal sequence, then for every n, n + max{ µ(e k ), µ(ê k ), (e k ), (ê k ) : 0 k n} 6π..4 An extension to Riesz bases Recall that {x k } is a Riesz basis for L (R) if there exists an isomorphism, U : L (R) L (R), called the orthogonalizer of {x k }, such that {Ux k } is an orthonormal basis for L (R). It then follows that, for every {a n } n=0 l, U a n a n x n n=0 n=0 U n=0 a n. (7) 8

9 One can adapt the results of the previous sections to Riesz bases. To start, note that the Rayleigh-Ritz technique leads to the following, cf. [0, Theorem XIII.3, page 8]: Lemma.7 Let H be a positive, self-adjoint, densely defined operator on L (R), and let {x k } be a Riesz basis for L (R) with orthonormalizer U. Then, for every n 0, n λ k (H) U n Hx k, x k. (8) PROOF. Let us take the notations of the proof of Corollary.. Write ϕ k = Ux k, it is then enough to notice that M = [ Hx k, x k ] = [ HU x k, U x k ] = [ U HU x k, x k ]. As U HU is a positive operator, that the Rayleigh-Ritz theorem gives But, λ k (U HU ) = n n Hx k, x k λ k (U HU ). sup ϕ 0,,ϕ k = sup inf ψ [ϕ 0,,ϕ k ], ψ inf ϕ 0,,ϕ k ψ [ϕ 0,,ϕ k ], ψ = sup ϕ 0,,ϕ k and, as U ψ U ψ, λ k (U HU ) U sup ϕ 0,, ϕ k inf ψ [U ϕ 0,,U ϕ k ], U ψ U HU ψ, ψ HU ψ, U ψ H ψ, ψ inf H ψ, ψ = ψ [ ϕ 0,, ϕ k ], ψ U λ k(h). Adapting the proofs of the previous section, we obtain the following corollary. Corollary.8 If {x k } is a Riesz basis for L (R) with orthonormalizer U then for all n, n ( (x k ) + ( x k ) + µ(x k ) + µ( x k ) ) (n + )(n + ) 4π U. 9

10 Thus, for every A > 0, there are at most 8πA U elements of the basis {x k } such that µ(e n ), µ(ê n ), (e n ), (ê n ) are all bounded by A. In particular, max{ µ(x k ), µ( x k ), (x k ), ( x k ) : 0 k n} n + U 6π. 3 Finite dimensional approximations, spherical codes and the Umbrella Theorem 3. Spherical codes Let K be either R or C, and let d be a fixed integer. We equip K d with the standard Euclidean scalar product and norm. We denote by S d the unit sphere of K d. Definition 3. Let A be a subset of {z K : z }. A spherical A-code is a finite subset V S d such that if u, v V and u v then u, v A. Let N K (A, d) denote the maximal cardinality of a spherical A-code. This notion has been introduced in [8] in the case K = R where upper-bounds on N R (A, d) have been obtained. These are important quantities in geometric combinatorics, and there is a large associated literature. Apart from [8], the results we use can all be found in [6]. Our prime interest is in the quantity NK(α, s N R ([ α, α], d), d) = N C ({z C : z α}, d), when K = R when K = C for α (0, ]. Of course N s R(α, d) N s C(α, d). Using the standard identification of C d with R d, namely identifying Z = (x + iy,..., x d + iy d ) C d with Z = (x, y,..., x d, y d ) R d, we have Z, Z R d = Re Z, Z C d. Thus N s C(α, d) N s R(α, d). In dimensions d = and d = one can compute the following values for N s K(α, d): NR(α, s ) = If 0 α < / then NR(α, s ) = If cos π α < cos π and 3 N then N s N N+ R(α, ) = N. In higher dimensions, one has the following result. 0

11 Lemma 3. If 0 α < d then N s K(α, d) = d. PROOF. An orthonormal basis of K d is a spherical [ α, α]-code so that NK(α, s d) d. For the converse, let α < /d and assume towards a contradiction that w 0,..., w d is a spherical [ α, α]-code. Indeed, let us show that d w 0,..., w d would be linearly independent in K d. Suppose that λ j w j = 0, and without loss of generality that λ j λ 0 for j =,..., d. Then λ 0 w 0 = d λ j w j, w 0 so that λ 0 λ 0 dα. As dα < we get that λ 0 = 0 and then j= λ j = 0 for all j. j=0 In general, it is difficult to compute N s K(α, k). A coarse estimate using volume counting proceeds as follows. Lemma 3.3 If 0 α < is fixed, then there exist constants 0 < a < a and 0 < C such that for all d C ea d N s K(α, d) Ce a d. Moreover, for α / one has N s K(α, d) 3 d if K = R, and N s K(α, d) 9 d if K = C. PROOF. The counting argument for the upper bound proceeds as follows. Let {w j } N n= be a spherical A-code, with A = [ α, α] or A = {z C : z α}. For j k, one has w j w k = w j + w k + Re w j, w k α. α So, the open balls B w j, α of center w j and radius are all α disjoint and included in the ball of center 0 and radius +. Therefore ( ) α hd/ Nc d c d + α where c d is the volume of the unit ball in K d, h = if K = R and h = if K = C. This gives the bound N ( + ) hd. α Note that for α / we get N 3 d if K = R and N 9 d if K = C. The lower bound too may be obtained by a volume counting argument, see [6]. hd

12 The work of Delsarte, Goethals, Seidel, [8] provides a method for obtaining more refined estimates on the size of spherical codes. For example, taking β = α in Example 4.5 of [8] shows that if α < d then N s R(α, d) ( α )d α d. (9) Equality can only occur for spherical { α, α}-codes. Also, note that if α = d, then α d d k α d dk+. 3. Approximations of orthonormal bases We now make a connection between the cardinality of spherical codes and projections of orthonormal bases. Let H be a Hilbert space over K and let Ψ = {ψ k } k= be an orthonormal basis for H. For an integer d, let P d be the orthogonal projection on the span of {ψ,..., ψ d }. For ε > 0, we say that an element f H is ε, d-approximable if f P d f H < ε, and define S ε,d to be the set of all of f H with f H = that are ε, d-approximable. We denote by A K (ε, d) the maximal cardinality of an orthonormal sequence in S ε,d. Example 3.4 Let {e j } n j= be the canonical basis for R n, and let {ψ j } n j= be an orthonormal basis for V, where V = span{(,,..., )}. Then e k P n e k = n holds for each k n, and hence A R ( n, n ) n. Our interest in spherical codes stems from the following result, cf. [9, Corollary ]. Proposition 3.5 If 0 < ε < / and α = ε, then A ε K (ε, d) NK(α, s d). PROOF. Let {ψ k } k= be an orthonormal basis for H, and let S ε,d and P d be as above. Let {f j } N j= H be an orthonormal set contained in S ε,d. For each d k =,..., N, j =,..., d, let a k,j = f k, ψ j and write P d f k := a k,j ψ j so that f k P d f k H < ε. Write v k = (a k,,..., a k,d ) K d then, for k l j=

13 v k, v l = P d f k, P d f l = P d f k f k + f k, P d f l f l + f l = P d f k f k, P d f l f l + P d f k f k, f l + f k, P d f l f l = P d f k f k, P d f l f l + P d f k f k, f l P d f l + f k P d f k, P d f l f l = f k P d f k, P d f l f l (0) since P d f k f k is orthogonal to P d f l. It follows from the Cauchy-Schwarz inequality that v k, v l ε. On the other hand, v k K d = P d f k H = ( f k H f k P d f k H )/ ( ε ) /. v k It follows that w k = satisfies, for k l, w k, w l = v k,v l v k ε v k K d K d v l, K d ε and {w k } N k= is a spherical [ α, α]-code in K d. Note that the proof only uses orthogonality in a mild way. Namely, if instead {f j } N j= H with f j H = satisfies f j, f k η for j k, then Equation (0) becomes v k, v l = f k P d f k, P d f l f l + f k, f( l, so that v k, v l ε ε + η, and the end of the proof shows that N NK s + η ) ε, d. In view of Proposition 3.5, it is natural ask the following question. Given α = ε, is there a converse inequality of the form N ε K(α, s d) CA K (ε, d ) with C > 0 an absolute constant and ε ε Cε, d d Cd? Note that for ε such that α < /d, we have A K (ε, d) = NK(α, s d) = d. 3.3 Prolate spheroïdal wave functions In order to obtain quantitative versions of Shapiro s theorems, we will make use of the prolate spheroïdal wave functions. For a detailed presentation on prolate spheroïdal wave functions see [,7,8]. Fix T, Ω > 0 and let {ψ n } n=0 be the associated prolate spheroidal wave functions, as defined in []. {ψ n } n=0 is an orthonormal basis for P W Ω {f L (R) : supp f [ Ω, Ω]}, and the ψ n are eigenfunctions of the differential operator L = (T x ) d dx x d dx Ω T x. As in the previous section, for an integer d 0, define P d to be the projection onto the span of ψ 0,..., ψ d, and for ε > 0 define S ε,d = {f L (R) : f =, f P d f < ε}. 3

14 For the remainder of the paper, the orthonormal basis used in the definitions of S ε,d, P d, and A K (ε, d), will always be chosen as the prolate spheroïdal wavefunctions. Note that these quantities depend on the choice of T, Ω. Finally, let { P T,Ω,ε = f L (R) : t >T f(t) dt ε and ξ >Ω f(ξ) dξ ε } and P T,ε = P T,T,ε. Theorem 3.6 (Landau-Pollak [8]) Let T, ε be positive constants and let d = 4T Ω +. Then, for every f P T,Ω,ε, f P d f 49ε f. In other words, P T,Ω,ε {f L (R) : f = } S 7ε,d. It follows that the first d = 4T + elements of the prolate spheroïdal basis well approximate P T,ε, and that P T,ε is essentially d-dimensional. 3.4 Generalized means and dispersions As an application of the results on prolate spheroidal wavefunctions and spherical codes, we shall address a more general version of the mean-dispersion theorem. Consider the following generalized means and variances. For p > and f L (R) with f =, we define the following associated p-variance p(f) = inf a R t a p f(t) dt. One can show that the infimum is actually a minimum and is attained for a unique a R that we call the p-mean µ p (f) = arg min a R t a p f(t) dt. As before, define the p-dispersion p (f) p(f). The proof of the Mean-Dispersion Theorem for p = via the Rayleigh-Ritz technique relied on the special relation (3) of means and dispersions with the Hermite operator. In general, beyond the case p =, such simple relations are not present and the techniques of Section are not so easily applicable. However, we shall show how to use the combinatorial techniques from the 4

15 beginning of this section to obtain a quantitative version of Theorem.3 for generalized means and dispersions. The following lemma is a modification of [9, Lemma ]. Lemma 3.7 Let A > 0 and p >. Suppose g L (R), g = satisfies µ p (g), µ p (ĝ), p (g), p (ĝ) A. Fix ε > 0, then g P A+(A/ε) /p,ε. This gives a simple proof of a strengthened version of Shapiro s Mean-Dispersion Theorem: Corollary 3.8 Let 0 < A, < p <, 0 < ε < /7, α = 49ε /( 49ε ), and set d = 4 ( A + (A/ε) /p) +. If {e k } N k= L (R) is an orthonormal set such that for all k N, µ p (e k ), µ p (ê k ), p (e k ), p (ê k ) A, then N N s C(α, d) N s R(α, d). PROOF. According to Lemma 3.7, e,..., e n are in P A+(A/ε) /p,ε. The definition of d and Theorem 3.6 show that {e j } N j= S 7ε,d. According to Proposition 3.5, N A C (7ε, d) N s C(α, d) N s R(α, d), where α = 49ε /( 49ε ). This approach does not, in general, give sharp results. For example, in the case p = the bound obtained by Corollary 3.8 is not as good as the one given in Section. To see this, assume that p = and A. Then 4A ( + /ε) d 5A ( + /ε). In order to apply the Delsarte, Goethals, Seidal bound (9) we will now chose ε so that α < which will then give that n 4d. Our d aim is thus to take d is as small as possible by chosing ε as large as possible. For this, let us first take ε /50 so that α 50ε. It is then enough that 50ε, that is 4A(+/ε) ε + ε 0. We may thus take ε = + 50A. 00A 5

16 Note that, as A, we get that ε + 50 < /50. This then gives n 0d 0A A ( = 0A + 00A + ) 50A + CA 4. In particular, the combinatorial methods allow one to take N = CA 4 in Corollary 3.8, whereas the sharp methods of Section.3 give N = 8πA, see Corollary The Quantitative Umbrella Theorem A second application of our method is a quantitative form of Shapiro s umbrella theorem. As with the mean-dispersion theorem, Shapiro s proof does not provide a bound on the number of elements in the sequence. As before, the combinatorial approach is well suited to this setting whereas the approach of Section is not easily applicable. Given f L (R) and ε > 0, define { C f (ε) = inf T 0 : } f(t) ε f. t >T If f is not identically zero then for all 0 < ε < one has 0 < C f (ε) <. Theorem 3.9 Let ϕ, ψ L (R) and M = min{ ϕ, ψ }. Fix 50M ε > 0, T > max{c ϕ (ε), C ψ (ε)}, and d = 4T +. If {e n } N n= is an orthonormal sequence in L (R) such that for all n N, and for almost all x, ξ R, e n (x) ϕ(x) and ê n (ξ) ψ(ξ), () then N N s C(50ε M, d) N s R(50ε M, d). () In particular, N is bounded by an absolute constant depending only on ϕ, ψ. PROOF. By (), T > max{c ϕ (ε), C ψ (ε)}, implies {e n } N n= P T,εM. According to Theorem 3.6, P T,εM S 7εM,d. It now follows from Proposition 3.5, 6

17 that N A C (7εM, d) N s C ( 49ε M ) 49ε M, d NC(50ε s M, d) NR(50ε s M, d). Let us give two applications where one may get an explicit upper bound by making a proper choice of ε in the proof above. Proposition 3.0 Let / < p and p C be fixed. If {e n } N n= L (R) is an orthonormal set such that for all n N, and for almost every x, ξ R, C C e n (x) and ê ( + x ) p n (ξ) ( + ξ ), p then ( ) 00 4 C p p 9 N 6 ( 400C p 4 ( 500C p ) p ) p 3 if / < p, if < p 3/, if 3/ < p. PROOF. If ϕ(x) = C, then M = ϕ (+ x ) p = C, and a computation for 0 < ε shows that C ϕ (ε) =. Let δ = δ(ε) = p 4 ε /(p ) ε 4/(p ) and α = α(ε) = 00C ε. Taking T = C p ϕ(ε) implies that d = 4T + δ(ε). If 0 < ε 50M, then Theorem 3.9 gives the bound N N s C (α(ε), δ(ε)). We shall chose ε differently for the various cases. Case. For the case / < p, take ε =, and use the exponential bound 50M given by Lemma 3.3 for NC(α(ε), s δ(ε)) to obtain the desired estimate. Case. For the case < p 3/, let ε 0 = ( p δ(ε 0 ). Note that α = δ < 0C δ, and also that ε 0 ) p (p ), α = α(ε 0 ), and δ =, since < p 3/. Thus the bound (9) yields N N s R(α, δ) = 4( α )δ 4δ. The desired estimate follows. Case 3. For the case 3/ < p, define ε = ( ) p p 50C p 3 and note that ε. 50M Since 3/ < p, taking ε < ε, α = α(ε), δ = δ(ε), implies that α(ε) < /δ(ε). Thus, by Lemma 3., N δ(ε) for all ε < ε. Hence, N δ(ε ), and the desired estimate follows. 50M 7

18 Note that in the case / < p, the upper bound in Proposition 3.0 approaches infinity as p approaches /. Indeed, we refer the reader to the counterexamples for p < / in [0,?]. The case p = / seems to be open as [0] need an extra logarithmic factor in their construction. For perspective in the case 3/ < p, if one takes C = C p = p, then the upper bound in Proposition 3.0 approaches 4 as p approaches infinity. Proposition 3. Let 0 < a and (a) /4 C be fixed. If {e n } N n=0 L (R) is an orthonormal set such that for all n and for almost every x, ξ R e n (x) Ce πa x and ê n (ξ) Ce πa ξ, then N + 8 { ( ) ( 50C πe π 50πC aπ max e πa/ ) } ln, ln. a /4 a 5/ e π PROOF. Let γ a (x) = Ce πa ξ and let C a (ε) = C γa (ε). First note that t >T γ a (t) dt = C e πa t dt = C ( + s )e πs t >T a T ds a + s C π( + at ) e πat, a while M = γ a = ( R C e πa t dt) / = C. In particular, γ (a) /4 a. Now for every T > 0, set ε(t ) = /4 π ( ) + at e πat, so that C a ε(t ) T. By Theorem 3.9, we get that N NR(50ε s (T )M, 8T + ), provided ε(t ). Let us first see what condition should be imposed on T to have ε(t ) 50M. Setting s = ( + at ), this condition is equivalent to se πs e π a /4 50M 50C. π Thus, it suffices to take s ln ( 50C ) πe π π a, and T ln ( 50C ) πe π /4 aπ a. /4 We will now further choose T large enough to have 50ε(T ) M <, so that 8T + Lemma 3. will imply N NR(50ε(T s ) M, 8T +) = 8T +. This time, the condition reads ( + at )( + 4T )e πat < a. Let r = a(4t + ). Thus, 50πC it suffices to take r e π r < a5/ e π It is enough to take r > 4 ln ( ) 50πC e πa/ 50πC e πa/ π a 5/ e, π and T > ln ( ) 50πC e πa/ aπ a 5/ e. π Combining the bounds for T from the previous two paragraphs yields N + 8 { ( ) ( 50C πe π 50πC aπ max e πa/ ) } ln, ln. a /4 a 5/ e π 8

19 A careful reading of the proof of the Umbrella Theorem shows the following: Proposition 3. Let 0 < C, and let p, q, p, q satisfy p + q = and p + q =. Let ϕ Lp (R) and ψ L p (R), and suppose that ϕ k L q (R) and ψ k L q (R) satisfy ϕ k q C, ψ k q C. There exists N such that, if {e k } L (R) is an orthonormal set which for all k and almost every x, ξ R satisfies e k (x) ϕ k (x) ϕ(x) and ê k (ξ) ψ k (ξ) ψ(ξ), then {e k } has at most N elements. As with previous results, a bound for N can be obtained in terms of spherical codes. The bound for N depends only on ϕ, ψ, C. Indeed, let ε > 0 and take T > 0 big enough to have Then t >T e k (t) dt t >T ( t >T ϕ k (t)ϕ(t) dt ϕ k (t) q dt C (ε p /C /p ) /p = ε. ) /q ( t >T t >T f(t) p dt ε p /C /p. ϕ(t) p dt ) /p A similar estimate holds for ê k and we conclude as in the proof of the Umbrella Theorem. 3.6 Angles in Riesz bases Let us now conclude this section with a few remarks on Riesz bases. Let {x k } be a Riesz basis for L (R) with orthogonalizer U and recall that, for every sequence {a n } n=0 l, U a n a n x n n=0 n=0 U n=0 a n. (3) Taking a n = δ n,k in (3) shows that U x k U. Then taking 9

20 a n = δ n,k + λδ n,l, k l and λ = t, t, t > 0 gives U ( + t ) x k + t x l + t Re x k, x l U ( + t ) thus Re x k, x l is ( min ( x k U )( x l U ), ( U xk U ) )( xl ) ( ) ( x k x l min U ) U U, U while taking λ = it, it, t > 0 gives the same bound for Im x k, x l. It follows that x k, x l C(U) x k x l C(U) U (4) where C(U) := min ( U U ), ( U ). U We may now adapt the proof of Proposition 3.5 to Riesz basis: Proposition 3.3 Let {ψ k ) k= be an orthonormal basis for L (R). Fix d 0 and let P d be the projection on the span of {ψ,..., ψ d }. Let {x k } k= be a Riesz basis ( for L (R) and let U be ) its orthogonalizer. Let ε > 0 be such that ε < min U, U C(U) U and let α = ε + C(U) U U ε. (5) If {x k } N k= satisfies x k P d x k < ε then N N s K(α, d). PROOF. Assume without loss of generality that x 0,..., x N satisfy x k P d x k < ε and let a k,j = x k, ψ j. Write v k = (a k,,..., a k,d ) K d then, the same computation as in (0), for k l v k, v l = x k P d x k, P d x l x l + x k, x l thus v k, v l ε + x k, x l. On the other hand v k = P d x k = ( x k x k P d f k ) / ( U ε ) / 0

21 It follows from (4) that w k = v k v k satisfies, for k l, w k, w l ε + C(U) U U ε and {w k } forms a spherical [ α, α]-code in K d. Note that the condition on ε implies that 0 < α <. Also note that if U is a near isometry in the sense that ( + β) U U + β then C(U) β(+β) and α (+β)ε +β(+β). In particular, if U is near enough to (+β) (+β)ε an isometry, meaning that β is small enough, then this α is comparable with the α of Proposition 3.5. As a consequence, we may then easily adapt the proof of results that relied on Proposition 3.5 to the statements about Riesz bases. For example, an Umbrella Theorem for Riesz bases reads as follows: Theorem 3.4 Let ϕ, ψ L (R) with ϕ, ψ. Let {f n } n= be a Riesz basis for L (R) with orthonormalizer U that is near enough to an isometry ( + β) U U + β with β small enough. Then there exists a constant N = N(ϕ, ψ, β) depending only on ϕ, ψ and β, such that the number of terms of the basis that satisfies f n (x) ϕ(x) and f n (ξ) ψ(ξ) for almost all x, ξ R is bounded by N. As with previous results, a bound on N can be given in terms of spherical codes. Acknowledgements A portion of this work was performed during the Erwin Schrödinger Institute (ESI) Special Semester on Modern methods of time-frequency analysis. The authors gratefully acknowledge ESI for its hospitality and financial support. The authors also thank Professor H.S. Shapiro for valuable comments related to the material. References [] G. Battle, Phase space localization theorem for ondelettes, J. Math. Phys., 30 (989),

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