On uncertainty principles in the finite dimensional setting

arXiv:0903.2923v1 [math.CA] 17 Mar 2009 ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING SAIFALLAH GHOBBER AND PHILIPPE JAMING Abstract. The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known “qualitative” uncertainty principles into more quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform. 1. Introduction The aim of this paper is to deal with uncertainty principles in finite dimensional settings. Usually, an uncertainty principle says that a function and its Fourier transform can not be both well concentrated. Of course, one needs to give a precise meaning to “well concentrated” and we refer to [15, 12] for numerous versions of the uncertainty principle for the Fourier transform in various settings. Our aim here is to present results of that flavour for unitary operators on Cd and then to apply those results to the discrete short-time Fourier transform. Before presenting our results, let us first introduce some notation. Let d be an integer and ℓ2d be Cd equipped with its standard norm denoted kakℓ2 or simply kak2 and the associated scalar product h·, ·i. More generally, for 0 < p < +∞, the ℓp -“norm” is defined by kakℓp = P 1/p d−1 p |a | . For a set E ⊂ {0, . . . , d − 1} we will write E c for its complementary, |E| j=0 j for the number of its elements. Further, for a = (a0 , . . . , ad−1 ) ∈ ℓ2d , we denote kakℓ2 (E) = P 1 2 2 . Finally, the support of a is defined as supp a = {j : a 6= 0} and we set |a | j j∈E j kakℓ0 = |supp a|. Our aim here is to deal with finite dimensional analogues of the uncertainty principle where concentration is measured in the following sense: Definition. Let T : ℓ2d → ℓ2d be a linear operator, S, Σ ⊂ {0, . . . , d − 1}. Then (S, Σ) is said to be a — weak annihilating pair (for T ) if supp a ⊂ S and supp T a ⊂ Σ implies that a = 0; — strong annihilating pair (for T ) if there exists a constant C(S, Σ) such that for every a ∈ ℓ2n
(1.1) kakℓ2 ≤ C(S, Σ) kakℓ2 (S c ) + kT akℓ2 (Σc ) . Of course, any strong annihilating pair is also a weak one. The corresponding notion for the Fourier transform has been extensively studied, and we refer to [15, 12] for more references. 1991 Mathematics Subject Classification. 42A68;42C20. Key words and phrases. Fourier transform, short-time Fourier transform, uncertainty principle. 1 2 SAIFALLAH GHOBBER AND PHILIPPE JAMING The advantage of the second notion over the first one is that it states that if the coordinates of a outside S and those of T a outside Σ are small, then a itself is small. It follows from a standard compactness argument (that we reproduce after Formula (2.5) below) that, in a finite dimensional setting, both notions are equivalent. However, this argument does not give any information on C(S, Σ). It is our aim here to modify an argument from [10] to obtain quantitative information on this constant in terms of S and Σ. We will restrict our attention to the case where T is invertible. Such an operator can be seen as a change of basis and we are thus looking for uncertainty principles of the following form: “A finite dimensional vector can not have coordinates concentrated in two different bases”. Such uncertainty principles have been known for some time. The first occurrence in the case of T being the discrete Fourier transform seems to be [22] and was rediscovered in [10]. Further results in that case can be found in [25, 23]. For more general changes of basis, we refer to [9, 11, 13]. Before going on with the results of this paper, let us mention its close connection with the Uniform Uncertainty Principle introduced by E. Candès and T. Tao in their seminal papers on compressed sensing [4, 5, 6, 7]: Definition. Let T : ℓ2d → ℓ2d be a unitary operator. Let s ≤ d be an integer and Ω ⊂ {0, . . . , d − 1}. Then (T, Ω, s) is said to have the Uniform Uncertainty Principle (also called the Restricted Isometry Property) if there exists δs ∈ (0, 1) such that, for every S ⊂ {0, . . . , d − 1} with |S| = s and for every a ∈ ℓ2d with supp a ⊂ S (1.2) (1 − δs )kak22 ≤ kT ak2ℓ2 (Ω) . We will call δs the Restricted Isometry Constant of (T, Ω, s). Note that, as T is unitary, (1.3) kT ak2ℓ2 (Ω) = kT ak22 − kT ak2ℓ2 (Ωc ) ≤ kak22 ≤ (1 + δs )kak22 . This inequality is sometimes included in the definition of the Uniform Uncertainty Principle and needed if T is not unitary. The purpose of this property was to show that, one may recover a from the knowledge of T a, under the restriction of a to be sufficiently sparse, that is |supp a| to be sufficiently small. Moreover, a may be reconstructed by ℓ1 -minimization. Let us mention how a recent result due to E. Candès [3] adapts in our case: Theorem 1.1 (Candès [3]). T : ℓ2d → ℓ2d be a unitary operator, let s ≤ d−1 2 be an integer and and Ω ⊂ {0, . . . , d − 1} and ε > 0. Assume that (T, Ω, 2s) satisfies the Uniform Uncertainty Principle with restricted √ √ isometry constant δ2s satisfying δ2s < 2 − 1 = ( 2 + 1)−1 . Then, for every a ∈ ℓ2d with |supp a| ≤ s, for every e ∈ ℓ2d with kek2 < ε, the solution a of
min{kãkℓ1 : ã ∈ Cd , T ã − T a + e 2 ≤ ε} satisfies √ 1 + ( 2 − 1)δ2s −1/2 4 √ √ s ka − a∗s k2 + ǫ ka − ak2 ≤ 2 1 − ( 2 + 1)δ2s 1 − ( 2 + 1)δ2s ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 3 where a∗s is a vector that minimizes ka − as k2 among all vectors as such that |supp as | ≤ s. In particular, if a has support of size s (thus a = a∗s ) and ε = 0, then a = a. In [3] the operator T is real, that is, it maps Rd → Rd . In order to adapt the result to complex operators, a stronger condition on δ2s needs to be imposed. On the other hand, we can take δs = 0 in (1.3), which allows for some gain, so that we are back to the same condition as in [3]. This follows from straight forward adaptations of the proofs in [3]. Let us now mention how the Uniform Uncertainty Principle (UUP) is linked to the notion of annihilating pairs. If (T, Ω, s) has the UUP with constant δs then, for every S of cardinality s, (S, Ωc ) is an annihilating pair. More precisely, a standard computation (see (2.7) where we reproduce the simple argument) shows that  

1 2 kak2 ≤ 1 + √ kakℓ2 (S c ) + kT akℓ2 (Ω) ≤ √ kakℓ2 (S c ) + kT akℓ2 (Ω) . 1 − δs 1 − δs Conversely, assume that Σ is such that, for every S such that |S| = s, (S, Σ) is a strong annihilating pair for T . Let C(Σ) = sup|S|=s C(S, Σ), then (T, Σc , s) satisfies the Uniform Uncertainty Principle with δs = 1 − C(Σ)−1 . Finally, we will apply our results to the discrete short-time Fourier transform. Let us describe these results in a slightly simplified setting. Let d be an integer, f, g ∈ ℓ2d , the short-time Fourier transform of f with window g is defined by d−1 1 X f (ℓ)g(ℓ − j)e2iπkℓ/d . Vg f (j, k) = √ d ℓ=0 We refer to e.g. [17, 16, 21] for various applications of the discrete short-time Fourier transform in signal processing. Our aim is to show that this transform satisfies an uncertainty principle. To do so, we adapt a method that was originally developed in [19, 20] and improved in [8, 14], that allows to transfer a result about strong annihilating pairs for the discrete Fourier transform into a similar result for its short-time version. A typical result will then be the following: Theorem 1.2. Let Σ be a subset of {0, . . . , d − 1}2 with |Σ| < d. then  1/2 √ X 2 2  |Vg f (j, k)|2  . kf k2 kgk2 ≤ 1 − |Σ|/d (j,k)∈Σ / This article is organized as follows: in the next section, we prove results about strong annihilating pairs for a change of basis. The following section deals with applications to the short-time Fourier transform. We devote the last section to a short conclusion. 2. The Uncertainty Principle for expansions in two bases. 2.1. Further notations on Hilbert spaces. Let Φ = {Φj }j=0,...,d−1 be a basis of Cd that is normalized i.e. kΦj k2 = 1 for all j. If d−1 X ai Φi . We will denote by kakℓp (Φ) = k(a0 , . . . , ad−1 )kℓp and a ∈ Cd , then we may write a = i=0 4 SAIFALLAH GHOBBER AND PHILIPPE JAMING suppΦ a = {i : ai 6= 0}. We also define kakℓ2 (Φ,E) in the obvious way when E is a subset of {0, . . . , d − 1}. When no confusion can arise, we simply write kakℓ2 (E) . Next, we will denote by Φ∗ = {Φ∗j }j=1,...,d the dual basis of Φ, that is the basis defined by ( 0 if j 6= k ∗ hΦj , Φk i = δj,k where δj,k is the Kronecker symbol, δj,k = . Every a ∈ Cd can 1 if j = k then be written as d−1 X a= a, Φ∗j Φj . j=0 Moreover, there exist two positive numbers α(Φ) and β(Φ), called the lower and upper Riesz bounds of Φ such that  1/2 d−1 X | a, Φ∗j |2  ≤ β(Φ)kak2 . (2.4) α(Φ)kak2 ≤  j=0 If Φ and Ψ are two normalized bases of Cd , we will define their coherence by M (Φ, ψ) = max 0≤j,k≤d−1 |hΦj , Ψk i|. 1 Obviously M (Φ, Ψ) ≤ 1 and, if Φ and Ψ are orthonormal bases, then M (Φ, Ψ) ≥ √ . Let d us recall that if M (Φ, Ψ) = √1 , then Φ and Ψ are said to be unbiased. A typical example of d a pair of unbiased bases is the standard basis and the Fourier basis of Cd , see Section 3.1. Let us recall that the Hilbert-Schmidt norm of a linear operator is the ℓ2d norm of its matrix in an orthonormal basis Φ: 1  2 d−1 X 2 kU kHS =  |hU Φi , Φj i|  . i,j=0 As is well known, this definition does not depend on the orthonormal basis and it controls the norm of U : ℓ2d → ℓ2d : kU kℓ2 →ℓ2 := d d max a∈Cd : kak2 =1 kU ak2 ≤ kU kHS . 2.2. The strong version of Elad and Bruckstein’s Uncertainty Principle. Let us start by giving a simple proof of a result of Elad and Bruckstein [11]. Lemma 2.1. Let Φ and Ψ be two normalized bases of Cd . Then, for every a ∈ Cd \ {0}, 1 kakℓ0 (Φ) kakℓ0 (Ψ) ≥  n o2 . β(Φ) ∗ , Ψ) min α(Ψ) M (Φ, Ψ∗ ), β(Ψ) M (Φ α(Φ) ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 5 In particular, kakℓ0 (Φ) + kakℓ0 (Ψ) ≥ min n 2 o. β(Φ) ∗ β(Ψ) ∗ α(Ψ) M (Φ, Ψ ), α(Φ) M (Φ , Ψ) Proof. The second statement immediately follows from the first one. The proof mimics the proof given in [25] for the Fourier basis. For a 6= 0 and j = 0, . . . , d − 1, | a, Ψ∗j | = d−1 X k=0 ha, Φ∗k i Φk , Ψ∗j ∗ ≤ ≤ M (Φ, Ψ )|suppΦ a| ≤ β(Φ)M (Φ, Ψ j,k=0,...,d−1 d−1 X 1/2 ∗ max Φk , Ψ∗j |ha, Φ∗k i|2 k=0 1/2 )kakℓ0 (Φ) kakℓ2 d−1 X !1/2 d−1 X k=0 |ha, Φ∗k i| !1/2 ≤ β(Φ) 1/2 M (Φ, Ψ∗ )kakℓ0 (Φ) α(Ψ) ≤ β(Φ) 1/2 1/2 M (Φ, Ψ∗ )kakℓ0 (Φ) kakℓ0 (Ψ) max |ha, Ψ∗k i|. k=0,...,d−1 α(Ψ) It follows that kakℓ0 (Φ) kakℓ0 (Ψ) ≥  k=0 |ha, Ψ∗k i|2 −2 β(Φ) ∗ . M (Φ, Ψ ) α(Ψ) Exchanging the roles of Φ and Ψ, we obtain the result.  Let us just note that if Φ and Ψ are two unbiased orthonormal bases, then the result reads |suppΦ a||suppΨ a| ≥ d. An other formulation would be the following: Let S and Σ are two subsets of {1, . . . , d} with |S||Σ| < d. If suppΦ a ⊂ S and suppΨ a ⊂ Σ then a = 0. In other words, if we denote by F the (unitary) operator that is defined by FΦi = Ψi , then (S, Σ) is an annihilating pair for F. When Φ is the standard basis and Ψ the Fourier basis, this result stems back to Matolcsi-Szücks [22] and Donoho-Stark [10]. It follows that (S, Σ) is also a strong annihilating pair, i.e. there exists a constant C = C(S, Σ, Φ, Ψ) such that, for every a ∈ Kd ,   (2.5) kak2 ≤ C kakℓ2 (Φ,S c ) + kakℓ2 (Ψ,Σc ) . Indeed, by homogeneity it is enough to prove (2.5) when kak2 = 1. But the unit sphere Sd = {a ∈ Cd : kak2 = 1} of Cd is compact and the map a → kakℓ2 (Φ,S c ) + kakℓ2 (Ψ,Σc ) is continuous, thus its minimum D over Sd is reached in some a0 . If this minimum were 0, then suppΦ a0 ⊂ S and suppΨ a0 ⊂ Σ thus a0 = 0 which would contradict ka0 k2 = 1. Thus D > 0 and (2.5) is satisfied with C = D−1 . However, this does not allow to obtain an estimate on the constant C. We will overcome this in the next theorem. 6 SAIFALLAH GHOBBER AND PHILIPPE JAMING Theorem 2.2. Let d be an integer. Let Φ and Ψ be two orthonormal bases of Cd and S, Σ be two subsets of 1 . Then, for every a ∈ Cd , {0, . . . , d − 1}. Assume that |S||Σ| < M (Φ, Ψ)2    1 kak2 ≤ 1 + kak + kak 2 c 2 c ℓ (Φ,S ) ℓ (Ψ,Σ ) . 1 − M (Φ, Ψ)(|S||Σ|)1/2 Remark : For comparison with the previous lemma, recall that as Φ, Ψ are orthonormal they are equal to their dual bases. Proof. Let U be the change of basis from Ψ to Φ, that is the linear operator defined by U Ψi = Φi . We will still denote by U its matrix in the basis Φi , so that U = [Ui,j ]1≤i,j≤d is given by Ui,j = hΦj , Ψi i. As U is unitary, U ∗ Φi = Ψi . P For a set E ⊂ {1, . . . , d} let PE be the projection PE a = j∈E ha, Φj iΦj . A direct computation then shows that kakℓ2 (Φ,S c ) = kPS c ak2 while  1/2  1/2 X X kakℓ2 (Ψ,E) =  |ha, Ψj i|2  =  |ha, U ∗ Φj i|2  j∈E  =  X j∈E j∈E 1/2 |hU a, Φj i|2  = kPE U ak2 .  = d X j=1 1/2 |hPE U a, Φj i|2  Assume first that a ∈ Cd is such that suppΦ a ⊂ S. Then kPΣ U ak2 = kPΣ U PS ak2 ≤ kPΣ U PS kℓ2 →ℓ2 kakℓ2 (Φ,S) . It follows that kakℓ2 (Ψ,Σc ) = kPΣc U ak2 ≥ kU ak2 − kPΣ U ak2 = kak2 − kPΣ U PS kℓ2 →ℓ2 kakℓ2 (Φ,S)   (2.6) = 1 − kPΣ U PS kℓ2 →ℓ2 kakℓ2 (Φ,S) . The last equality comes from the assumption suppΦ x ⊂ S which implies kak2 = kakℓ2 (Φ,S) . Note that, if we are able to prove that kPΣ U PS kℓ2 →ℓ2 < 1, then this inequality implies that (S, Σ) is an annihilating pair. The following computation allows to estimate the constant C(S, Σ) appearing in the definition of a strong annihilating pair: write D = −1  then, for a ∈ Cd , 1 − kPΣ U PS kℓ2 →ℓ2 (2.7) kak2 = kPS ak2 + kPS c ak2 ≤ DkPΣc U PS ak2 + kPS c ak2 = DkPΣc U (a − PS c a)k2 + kPS c ak2 ≤ DkPΣc U ak2 + DkU PS c ak2 + kPS c ak2 since kPΣc xk2 ≤ kxk2 for every x ∈ Cd . Now, as U is unitary, we get
kak2 ≤ DkPΣ U ak2 + 1 + D kPS c ak2 ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 7 which immediately gives an estimate of the desired form with −1  . C(S, Σ, Φ, Ψ) = 1 + 1 − kPΣ U PS kℓ2 →ℓ2 It remains to give an upper bound on kPΣ U PS kℓ2 →ℓ2 : 1/2  d d X X |hΦi , PΣ U PS Φj i|2  = kPΣ U PS k 2 2 ≤ kPΣ U PS k HS ℓ →ℓ i=1 j=1  (2.8) =  XX i∈Σ j∈S 1/2 |hΦi , U Φj i|2  ≤ M (Φ, Ψ)(|S||Σ|)1/2 which completes the proof of the theorem.  Remark : Let Φ = {Φj }j=0,...,d−1 and Ψ = {Ψj }j=0,...,d−1 be two orthonormal bases of Cd and let U be the operator defined by U Ψj = Φj for j = 0, . . . , d − 1. 1 then (U, s, Σc ) satisfy the uniform It follows from (2.8) and (2.6) that, if s|Σ| < M (Φ, Ψ)2 uncertainty principle with restriced isometry constant δs ≤ M (Φ, Ψ)2 s|Σ|. As a consequence, √ −2 c Theorem 1.1 applies as soon as s|Σ| ≤ 2−1 2 M (Φ, Ψ) . Writing Ω = Σ , we may deduce from √ 2−1 , then any a such that kakℓ0 (Φ) ≤ s can be recovered as this that, if s ≤ 2(d − |Ω|)|M (Φ, Ψ)2 the unique solution of min{kãkℓ1 : hã, Ψj i = ha, Ψj i for every j ∈ Ω}. Earlier results in that spirit may be found in [9, 11, 13]. Definition. Let C > 0 and α > 1/2. We will say that a ∈ Cd is (C, α)-compressible in the basis Φ if the √ C j-th biggest coefficient |ha, Φi|∗ (j) of a in the basis Φ satisfies |ha, Φi|∗ (j) ≤ 2α − 1 α kak. j In order to illustrate our main theorem, let us show that a vector can not be too compressible in two different bases. We will restrict to a simple enough case, the proof being easy to adapt to more general settings: Corollary 2.3. Let Φ and Ψ be two unbiased orthonormal bases of Cd . Let d ≥ 9, C > 0 and α > 1/2 be √ 1 ( d − 3)α− 2 √ . Then the only vector a that is (C, α)-compressible in both bases such that C < 4 d is 0. Proof. Let a 6= 0 and assume that a is (C, α)-compressible in both bases. Without loss of generality, we may assume that kak2 = 1. 8 SAIFALLAH GHOBBER AND PHILIPPE JAMING Let σ = σΦ be a permutation such that   |ha, Φσ(j) i| is non-increasing. For k = 1≤j≤d 0, . . . , d − 1 define Sk = {σφ (0), . . . , σΦ (k)}, the set of the k + 1 biggest coefficients of a in the basis Φ. Then d−1 X X kak2ℓ2 (Φ,S c ) = |ha, Φj i|2 = |ha, Φσ(j) i|2 k j=k+1 j ∈S / k ≤ (2α − 1)C 2 d−1 X j=k+1 j −2α ≤ (2α − 1)C 2 Z k +∞ C2 dx = . x2α k2α−1 C C It follows that kakℓ2 (Φ,Skc ) ≤ α− 1 . In a similar way, we get kakℓ2 (Ψ,Σc ) ≤ 1 where Σk is k 2 k k α− 2 the set of the k + 1 biggest coefficients of a in the basis Ψ. √ Let us now apply Theorem 2.2 with S = Sk and Σ = Σk . Then, as long as k + 1 < d, k + 1  α− 1 2C 2 1 √ 1 − × 1≤ k 2. . In other words, C ≥ α− 12 √ 4 1 − k+1 d k d √ Assume now√that d ≥ 9 and chose k = [ d] − 2 (where [x] is the largest integer less than x) so that k < d − 1. It follows that ! √ √ 1 √ [ d] − 1 ( d − 3)α− 2 1 α− 12 √ 1− √ ≥ ([ d] − 2) C≥ 4 d 4 d which completes the proof.  Remark : — This corollary may be seen as a discrete analogue of Hardy’s Uncertainty Principle which states that an L2 (R) function and its Fourier transform can not both decrease too fast (see [15, 12]). — The above proof also works if the bases are not unbiased, in which case the condition on C has to be replaced by α−1/2  1 M (Φ, Ψ) . −3 C< 4 M (Φ, Ψ) — Let Φ be an orthonormal basis of Cd and a ∈ Cd with kak = 1 and 0 ≤ p < 2. From Bienaymé-Tchebichev, we get that, for λ ≥ 0, |{j : |ha, Φj i| ≥ λ}| ≤ kxkpℓp (Φ) λ−p . Applying this to λ = |ha, Φi|∗ (k) we get k ≤ |{j : |ha, Φj i| ≥ |ha, Φi|∗ (k)}| ≤ kakpℓp (Φ) (|ha, Φi|∗ (k))−p thus ∗ kakℓp (Φ) r 2 −1 p r  p kakℓp (Φ) k−1/p . p−2 |ha, Φi| (k) ≤ = k1/p  r 1 p kakℓp (Φ) , -compressible in Φ. It follows that a is 2−p p This shows that a vector can not have coefficients in two bases with too small ℓp -norm. ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 9 2.3. Results on annihilating pairs using probability techniques. So far, we have only used deterministic techniques, which lead to rather weak results. In this section, we will recall some results that may be obtained using probability methods. First, let us describe a model of random subsets of cardinality k. Let k ≤ d be an integer and let δ0 , . . . , δd−1 be d independent random variables take take the value 1 with probability k/n and 0 with probability 1 − k/n. We then define the random subset of cardinality k, Ω ⊂ {0, . . . , d − 1} by Ω = {i : δi = 1}. The term “of cardinality k” is justified by the fact P that the average cardinality of Ω is k (which is immediate once one write 1Ω = d−1 j=0 δj 1j ). Moreover, one has the following standard estimate (see e.g. [1, Theorems A.1.12 and A.1.13] or [18]):   k ≤ 2e−k/10 . P |Ω − k| ≥ 2 Then Rudelson-Vershynin [24], (improving a result of Candès-Tao) proved the following theorem: Theorem 2.4 (Rudelson-Vershynin [24]). There exist two absolute constants C, c such that the following holds: let Φ = {Φ0 , . . . , Φd−1 } and Ψ = {Ψ0 , . . . , Ψd−1 } be two unbiased orthonormal bases of Cd and let T : ℓ2d → ℓ2d be defined by T ψj = Φj for j = 0, . . . , d − 1. Let 0 < η < 1, t > 1 be real numbers and s ≤ d be an integer. Let k be an integer such that (2.9) k ≃ (Cts log d) log(Cts log d) log2 s. Then, with probability at least 1 − 7e−c(1−η)t , a random set Ω of cardinality k satisfies √ √ k − tk ≤ |Ω| ≤ k + tk and (T, Ω, s) satisfies the Uniform Uncertainty Principle with Restricted Isometry Constant δs ≤ 1 − η. In particular, for any S ⊂ {0, . . . , d} with |S| ≤ s, for every a ∈ ℓ2d ,  2 (2.10) kakℓ2 ≤ √ kakℓ2 (Φ,S c ) + kakℓ2 (Ψ,Ω) . η The parameter η is not present in their statement, but it can be obtained by straightforward modification of their proof. log d d −κ(1−η) Taking s = 5 , t = 2C we obtain k ≃ d/2. Thus, with probability ≥ 1 − 5d log d (κ some universal constant) Ω has cardinal |Ω| = d/2 + O(d1/2 log1/2 d) and every set S with cardinal |S| ≤ logd5 d and Ωc form a strong annihilating pair in the sense of (2.10). An other question that one may ask is the following. Given a set Σ, does there exist a “large” set S such that (S, Σ) is an annihilating pair? In order to answer this question, let us recall that Bourgain-Tzafriri [2] proved the following: Theorem 2.5 (Bourgain-Tzafriri, [2]). If T : ℓ2n → ℓ2n is such that kT ei k2 = 1 for i = 0, . . . , n − 1 (where the ei ’s stand for the such standard basis of ℓ2n ), then there exists a set σ ⊂ {0, . . . , n − 1} with |σ| ≥ 240kT nk2 that, for every a = (aj )j=0,...,n−1 with support in σ such that kT ak ≥ 1 12 kak. ℓ2 →ℓ2 10 SAIFALLAH GHOBBER AND PHILIPPE JAMING The values of the numerical constants where given in [18]. We may apply this theorem in the following way: consider two mutually unbiased orthonormal bases Φ = {φj } and Ψ = {ψj } of ℓ2d and let S, Ω ⊂ {0, . . . , d − 1} be two sets with |S| = |Ω| = n and enumerate them: r S = {j0 , . . . , jn−1 } and Ω = {ω0 , . . . , ωn−1 }. Let T be d ψω for k = 0, . . . , n − 1. Then T satisfies the hypothesis the operator defined by T φjk = n k d of Bourgain-Tzafriri’s Theorem and kT k2 ≤ . Thus there exists σ ⊂ S with |σ| ≥ n2 /240d n such that, for every a ∈ ℓ2n with support in σ, r 1 n kakℓ2 (σ) . kakℓ2 (Ψ,Ω) ≥ 12 d From which we immediately deduce the following (where Σ = Ωc ): Proposition 2.6. Let Φ and Ψ be two mutually unbiased bases of ℓ2d and let S, Σ ⊂ {0, . . . , d − 1} be two sets (d − |Σ|)2 and, for every a ∈ ℓ2d , with |S| + |Σ| = d. Then there exists σ ⊂ S such that |σ| ≥ 240d
13 (2.11) kakℓ2 ≤ p kakℓ2 (Φ,σc ) + kakℓ2 (Ψ,Σc ) . 1 − |Σ|/d √ Of course, this proposition only makes sense when |Σ| ≤ d − 240d otherwise there is no guarantee to have σ 6= ∅. We may thus rewrite (2.11) as
kakℓ2 ≤ 4d1/4 kakℓ2 (Φ,σc ) + kakℓ2 (Ψ,Σc ) . 3. The uncertainty principle for the discrete short-time Fourier transform 3.1. Finite Abelian groups. In this section, we recall some notations on the Fourier transform on finite Abelian groups. Results stated here may be found in [26] and (with slightly modified notations) in [21]. Throughout the remaining of this paper, we will denote by G a finite Abelian group for which the group law will be denoted additively. The identity element of G is denoted by 0. The dual group of characters Ĝ of G is the set of homomorphisms ξ ∈ Ĝ which map G into the multiplicative group S1 = {z ∈ C : |z| = 1}. The set Ĝ is an Abelian group under pointwise multiplication and, as is customary, we shall write this commutative group operation additively. Note that G is isomorphic to Ĝ, in particular |G| = |Ĝ|. Further, b̂ Pontryagin duality implies that G can be canonically identified with G, a fact which is emphasized by writing hξ, xi = ξ(x). Note that, as group operations are written additively, h−ξ, xi = hξ, −xi = hξ, xi. The Fourier transform FG f = fˆ ∈ CĜ of f ∈ CG is given by 1 X f (x)hξ, xi, ξ ∈ Ĝ. fˆ(ξ) = |G|1/2 x∈G ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 11 The transform is unitary : kfˆk2 = kf k2 and the inversion formula for the Fourier transform allows us to reconstruct the original function from its Fourier transform. Namely, for f ∈ Ĝ we have 1 X ˆ f (x) = FGb [fˆ ](x) = f (ξ)hξ, xi, x ∈ G. |G|1/2 ξ∈Ĝ Moreover, as the normalized characters {|G|−1/2 ξ}ξ∈Ĝ form an orthonormal basis of CG that is unbiased with the standard basis we can reformulate Theorem 2.2 as follows: Strong Uncertainty Principle on Finite Abelian Groups. Let G be a finite Abelian group and let S ⊂ G and Σ ⊂ Ĝ be such that |S||Σ| < |G|. Then, for every f ∈ CG ,  1/2  !1/2  X 2   X 2  kf k2 ≤ |fˆ(ξ)|2   . |f (x)| + (3.12)  1/2 1 − (|S||Σ|/|G|) x∈S / ξ ∈Σ / The corresponding “weak” Uncertainty Principle appeared for the first time in [22] and has been re-discovered several times, including [10]. Note also that one may improve this results by chosing the sets S, Σ randomly ([28] for the cyclic groups) or when better information on the group is taken into account ([23, 25] for weak versions of this theorem, a proper estimate of the constants appearing in the corresponding strong versions remaining open). For any x ∈ G, we define the translation operator Tx as the unitary operator on CG given by Tx f (y) = f (y − x), y ∈ G. Similarly, we define the modulation operator Mξ . for ξ ∈ Ĝ as the unitary operator defined by Mξ f = f · ξ, where here and in the following f · g denotes the d ˆ pointwise product of f, g ∈ CG . Since M ξ f = Tξ f , we refer to Mξ also as a frequency shift ˆ operator. Note also that Td x f = M−x f We denote by π(λ) = Mξ Tx , λ = (x, ξ) ∈ G × Ĝ the time-frequency shift operators. Note that these are unitary operators. The short-time Fourier transformation VgG : C G → CG×Ĝ with respect to the window g ∈ CG \ {0} is given by VgG f (x, ξ) = 1 X 1 hf, π(x, ξ)gi = f (y)g(y − x) hξ, yi = FG [f · Tx g](ξ) |G|1/2 |G|1/2 y∈G where f ∈ CG . The inversion formula for the short-time Fourier transform is f (y) = 1 |G|1/2 kgk22 X (x,ξ)∈G×Ĝ VgG f (x, ξ)g(y − x)hξ, yi. Further, kVg k2 = kf k2 kgk2 , in particular Vg f = 0 if and only if either f = 0 or g = 0. Finally, let us note that a simple computation shows that (3.13) G Vπ(b,v)g π(a, u)f (x, ξ) = hu − v − ξ, aihv, xiVgG f (x − a + b, ξ − u + v). 12 SAIFALLAH GHOBBER AND PHILIPPE JAMING 3.2. The symmetry lemma. Let us first note that the short-time Fourier transform on Ĝ is defined by VγĜ ϕ(ξ, x) = 1 |Ĝ|1/2 hϕ, Mx Tξ γi = 1 |Ĝ|1/2 X η∈Ĝ ϕ(η)γ(η − ξ) η(x). This is linked to V G in the following way: VgG f (x, ξ) = = 1 1 F G f, F G [Mξ Tx g] hf, π(x, ξ)gi = |G|1/2 |G|1/2 1 |Ĝ|1/2 F G f, Tξ M−x F G g = hξ, xi |Ĝ|1/2 F G f, M−x Tξ F G g , so that VgG f (x, ξ) = hξ, xiVĝĜ fˆ(ξ, −x). (3.14) Lemma 3.1. Let f, g, h, k ∈ CG . Then, for every u ∈ G and every η ∈ Ĝ, FG×Ĝ [VgG f VhG k](η, u) = VkG f (−u, η)VhG g(−u, η). Proof. First note that 1 FG×Ĝ [VgG f VhG k](η, u) = |G|1/2 |Ĝ|1/2 1 = |G|1/2 |Ĝ|1/2 XX x∈G ξ∈Ĝ XX x∈G ξ∈Ĝ VgG f (x, ξ) VhG k(x, ξ) hη, xi hξ, ui VgG f (x, ξ) V Ĝ k̂(ξ, −x) hη, xi hξ, u − xi ĥ with (3.14). Using the definition of the short-time Fourier transform, this is further equal to 1 XXXX |Ĝ||G| x∈G (3.15) = ξ∈Ĝ y∈G ζ∈Ĝ 1 f (y)g(y − x) hξ, yi k̂(ζ)ĥ(ζ − ξ)hζ, −xi hη, xi hξ, u + xi XXXX |Ĝ||G| x∈G ξ∈Ĝ y∈G ζ∈Ĝ f (y)g(y − x) k̂(ζ)ĥ(ζ − ξ) hη + ζ, xi hξ, u − x + yi We will now invert the orders of summation. First 1 X 1 X ĥ(ζ − ξ)hξ, u − x + yi = ĥ(ξ + ζ)hξ, u − x + yi |Ĝ|1/2 |Ĝ|1/2 ξ∈Ĝ ξ∈Ĝ = 1 |Ĝ|1/2 X ξ∈Ĝ \ M −ζ h(ξ)hξ, u − x + yi = [M−ζ h](u − x + y) = hζ, xih−ζ, u + yih(u − x + y). ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 13 Then 1 |Ĝ|1/2 |G|1/2 X x∈G g(y − x) hζ + η, xi X ξ∈Ĝ ĥ(ζ − ξ)hξ, u − x + yi hζ, u + yi X g(y − x) hη, xih(u − x + y) = |G|1/2 x∈G = hζ, u + yi X g(z)h(z + u)hη, zi hη, yi |G|1/2 z∈G = hζ, ui hη + ζ, yi VhG g(−u, η). It follows that 1 |Ĝ||G|1/2 X k̂(ζ) X x∈G ζ∈Ĝ = = g(y − x) hζ + η, xi 1 |Ĝ|1/2 X ζ∈Ĝ X ξ∈Ĝ ĥ(ζ − ξ)hξ, u − x + yi k̂(ζ)hζ, ui hη + ζ, yi VhG g(−u, η) hη, yi VhG g(−u, η) X |Ĝ|1/2 k̂(ζ)hζ, y + ui ζ∈Ĝ = hη, yi VhG g(−u, η) k(y + u). Finally, it remains to take the sum in the y-variable in 3.15 to obtain FG×Ĝ [VgG f VhG k](η, u) = VkG f (−u, η)VhG g(−u, η). as announced.  Remark : The short-time Fourier transform can be defined on any locally Abelian group G and its dual Ĝ as Z (x, ξ) ∈ G × Ĝ f (t)g(t − x)hξ, ti dν(t), Vg f (x, ξ) = G where dνG is the Haar measure on G. All results in this section go through to this more general context, the inversions of order of integrations being easily justified. In the case G = Rd , this lemma was given independently in [19, 20]. 3.3. The Uncertainty Principle for the short-time Fourier transform. We will conclude this section with the following theorem that allows to transfer results about strong annihilating pairs in G × Ĝ to Uncertainty Principles for the short-time Fourier trnsform. \ Theorem 3.2. Let Σ ⊂ G × Ĝ and Σ̃ = {(ξ, −x) : (x, ξ) ∈ Σ} ⊂ G × Ĝ = Ĝ × G. Assume that (Σ, Σ̃) is strong annihilating pair in G × Ĝ, i.e. that there is a constant C(Σ) such that, 14 SAIFALLAH GHOBBER AND PHILIPPE JAMING for every F ∈ CG×Ĝ ,  kF k22 ≤ C(Σ)  X (x,ξ)∈Σ / then, for every f, g ∈ CG , |F (x, ξ)|2 + kf k22 kgk22 ≤ 2C(Σ) X X (x,ξ)∈ / Σ̃ (x,ξ)∈Σ /  |FG×Ĝ F (ξ, x)|2  |VgG f (x, ξ)|2 . Proof. We will adapt the proof in the case G = Rd given in [8] to our situation. Let us fix f, g ∈ CG . We will only use Lemma 3.1 in a simple form: for a ∈ G, η ∈ Ĝ define the function Fa,η on G × Ĝ by Fa,η (x, ξ) = hξ − η, aiVgG f (x − a, ξ − η)VfG g(x, ξ). G Note that Fa,η (x, ξ) = VgG π(a, η)f Vπ(a,η)f g so that then FG×Ĝ Fa,η (ξ, x) = Fa,η (−x, ξ). It follows that   X X kFa,η k22 ≤ C(Σ)  |Fa,η (x, ξ)|2 + |Fa,η (−x, ξ)|2  (x,ξ)∈Σ / = 2C(Σ) X (x,ξ)∈Σ / (x,ξ)∈ / Σ̃ |VgG f (x − a, ξ − η)|2 |VfG g(x, ξ)|2 . Finally, summing this inequality over (a, η) ∈ G × Ĝ gives X |VfG g(x, ξ)|2 kf k42 kgk42 ≤ 2C(Σ)kf k22 kgk22 (x,ξ)∈Σ / which completes the proof.  Combining this result with (3.12) we immediately get the following: Corollary 3.3. Let Σ ⊂ G × Ĝ with |Σ| < |G|. Then, for every f, g ∈ CG , X 8 |VfG g(x, ξ)|2 . kf k22 kgk22 ≤ (1 − |Σ|/|G|)2 (x,ξ)∈Σ / The corresponding weak annihilating property for Σ was obtained by F. Krahmer, G. E. Pfander and P. Rashkov [21]. Finally, using the fact that two random events A and B that each occur with probability ≥ 1 − α, jointly occur with probability ≥ 1 − 2α, we deduce the following from Theorem 2.4: Theorem 3.4. There exist two absolute constants C, c such that the following holds: Let 0 < η < 1, t > 1 be real numbers and s ≤ d be an integer. Let k be an integer such that k ≃ (Cts log d) log(Cts log d) log2 s. ON UNCERTAINTY PRINCIPLES IN THE FINITE DIMENSIONAL SETTING 15 Then, with probability at least 1 − 14e−c(1−η)t , a random set Ω of cardinality k satisfies √ √ k − tk ≤ |Ω| ≤ k + tk and, for any S ⊂ {0, . . . , d} with |S| ≤ s, for every f, g ∈ ℓ2d ,  1/2 √ X 2 2 |VfG g(x, ξ)|2  . 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Département Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, Campus Universitaire, 1060 Tunis, Tunisie E-mail address: [email protected] P.J. and S.G Université d’Orléans, Faculté des Sciences, MAPMO - Fédération Denis Poisson, BP 6759, F 45067 Orléans Cedex 2, France, Tel:+33 (0) 238 494 908, Fax:+33 (0) 238 417 205 E-mail address: [email protected]