The uncertainty principle for functions with sparse support and spectrum

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The uncertainty principle for functions with sparse support and spectrum

Rodolfo A. Ríos Zertuche R. Z.

U.U.D.M. Project Report 2005:7

Examensarbete i matematik, 20 poäng Handledare och examinator: Burglind Juhl-Jöricke

Juni 2005

Department of Mathematics

Uppsala University

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The Uncertainty Principle for functions with sparse support and spectrum

Rodolfo A. R´ıos Zertuche R. Z.

May 25, 2005

Abstract

The mathematical formulation of Heisenberg’s Uncertainty Princi- ple is a statement in the spirit of the following mathematical principle:

A function and its Fourier transform cannot be “too small” simultaneously unless vanishing identically. One of the forms of this mathematical principle is a statement that a non-zero function and its Fourier transform cannot be concentrated simultaneously on sparse sets.

We review and compare the versions of the Uncertainty Principle presented by Shubin, Vakilian, Wolff and Kovrizhkin. The spirit of the statements in both papers is the following: if the sets E and F satisfy certain sparseness conditions, then one gets, for some C > 0,kfk2²≤ C(kfk_L²2(E^c)+kfk_L²2(F^c)), for all f ∈ L²(R^d). Here, E^c = R^d\E.

Kovrizhkin’s version links the result of Shubin, Vakilian and Wolff, and the Logvinenko-Sereda theorem. We give alternative characteri- zations of the hypotheses in Kovrizhkin’s version, using the Legendre transform.

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1 Introductory remarks

1.1 Introduction

The term ‘Uncertainty Principle’ is borrowed from physics, where a function can be interpreted as the wave function of a particle (whose squared modulus is usually interpreted as the probability density of the position of the particle), while the Fourier transform is interpreted as the decomposition of the wave function into (infinitesimal) components of different wavelengths.

The momentum of a wave is linearly related to its frequency, so the squared modulus of the Fourier transform is closely related to the probability density of the momentum. Heisenberg’s famous statement of the Uncertainty Principle is the following:

The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.

Its mathematical formulation is a statement in the spirit of the following mathematical principle: A function and its Fourier transform cannot be

“too small” simultaneously, unless vanishing identically. There exist various realizations of the mathematical principle, often having direct applications in mathematical physics, the above mentioned form of the Uncertainty Prin- ciple being one of them.

One of the forms of the Uncertainty Principle in harmonic analysis is a statement that a function and its Fourier transform cannot be concentrated simultaneously in small sets. There are several versions of such statements, for different function spaces and different definitions of the smallness of the sets.

We give some examples. For a subset E of R^d(d a natural number), we denote E^c = R^d\E. Nazarov’s improvement of the Amrein-Berthier theorem says that, for sets E, F of finite measure there exists a constant C > 0 such

that Z

|f|²≤ C

Z

E^c

|f|²+ Z

F^c

| bf|

(1.1.1) holds for all f ∈ L²(R^d). As a consequence, the Amrein-Berthier theorem is obtained: if f and bf are supported on sets of finite measure, f = 0 almost everywhere.

Similarly, in the Logvinenko-Sereda theorem E^c is taken to be relatively dense, in the sense that for some ball B and some 0≤ γ ≤ 1, |E^c∩(B +x)| ≥ γ|B| holds for all x ∈ R^d, and F is taken to be compact. Then, there is a constant C > 0 such that if supp bf ⊂ F ,

Z

|f|²≤ C Z

E^c|f|². For many examples, the reader is referred to [1].

3

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4 1.3. Overview of the results

In this report, a couple of relatively new versions of this principle, which have found some applications in physics, are reviewed and discussed from the analytical point of view.

1.2 Some notations

For a measurable set S ⊂ R^d, we will often denote by |S| its Lebesgue measure and by χS its characteristic function. L²(S) will denote the space of square integrable functions f : S→ C, equipped with the norm k · k2(the square root of the integral of the square of the absolute value), while the norm k · k1 will denote, as usual, the integral of the absolute value. S(R^d) will denote the space of Schwartz functions on R^d.

We will take the Fourier transform of a function f ∈ L²(R^d) to be defined by bf (y) =R f (x) e^−2πixydx, with inverse ˇf (y) =R f (x) e^2πixydx. The restriction of a function f to a set S will be denoted f|S.

We will denote by B(x, r) the open ball with center at x ∈ R^d and radius r > 0. S^d⁻¹ ⊂ R^dis the unit sphere. R⁺ will denote the positive real numbers.

When comparing two real variables a and b which may change indepen- dently, we will write a . b if there is a constant C > 0 not depending on a and b for which a≤ Cb. a ∼ b will mean a . b and b . a.

1.3 Overview of the results

The main theorems presented here are the Uncertainty Principle found by C.

Shubin, R. Vakilian, and T. Wolff and published in 1998, and Kovrizhkin’s 2003 generalization of this result. The reader is referred to Sections 3 and 4.1 for the precise statements of the theorems.

In both cases, the smallness of the sets is defined as follows. Given ε > 0, a set A is defined to be ε-thin with respect to a function ρ : R⁺ → R⁺ if, for all x∈ R^d,

|A ∩ B(x, ρ(|x|))| ≤ ε|B(x, ρ(|x|))| .

In the versions of the Uncertainty Principle examined here, ρ will always tend to zero as|x| → ∞. Therefore, the above mentioned condition implies not only that the set A^c is relatively dense in R^d, but in addition, the relative density of A^c in the mentioned balls (of small size if|x| is big) is uniformly bounded from below.

In the case of the Shubin-Vakilian-Wolff theorem, ρ is given by ρ(|x|) = min(1,|x|⁻¹). The theorem guarantees the existence of some ε > 0 such that for all sets E, F that are ε-thin with respect to this ρ, there exists a constant C > 0 such that equation (1.1.1) holds for all f ∈ L²(R^d).

Kovrizhkin’s generalization of this result allows greater freedom in the selection of the sets E, F , which are now required to be ε-thin with respect

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1.4. Acknowledgments 5

to two (probably different) functions ρ₁ respectively ρ₂ which satisfy C₁

ρ1

C2

ρ2(t)

≥ t for large t > 0.

The same conclusion follows as for the Shubin-Vakilian-Wolff Theorem.

In Section 4.2, the condition on the ρ’s is found to be equivalent, in many cases, to the following expression involving the Legendre transform,L.

(L Z 1

˜ ρ1

)⁰≤ 1 ρ2

. Here, ˜ρ₁(t) = ρ₁(C₂t)/C₁.

Our approach for the exposition is the following. In Section 2, the general outline of the proofs is described, some lemmas are developed which are of common use to both proofs, and even a small well-known result is proved, which has been included here since it seems to motivate the development of the other proofs.

In Section 3, the Shubin-Vakilian-Wolff theorem is stated and proved.

Since Kovrizhkin’s work sprang from this proof, and since this seems to be a more pedagogical exposition, a reading of this section is taken as a prereq- uisite for Section 4. The later provides the proof of Kovrizhkin’s theorem as well as a discussion of the hypotheses. In Section 4.8, the connection is given between this theorem and the Logvinenko-Sereda theorem.

Finally, the known sharpness results for these theorems are discussed in Section 5.

1.4 Acknowledgments

The author would like to thank Burglind Juhl-J¨oricke for her guidance and continuous encouragement, which were essential for the development and completion of this paper.

“Also, I want to acknowledge the life-long support I have had from my parents, and the efforts of all those who have been my teachers in the past, which have all somehow influenced this work,” he remarks.

This paper was written during a one-year stay at Uppsala Universitet, Sweden. The author was partly supported by the University of Guanajuato and the Center for Mathematical Research (Cimat, A.C.), Mexico.

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2 Some preparations

There are some common points in the proofs of the Shubin-Vakilian-Wolff and Kovrizhkin theorems. In this section, we break them apart, hoping that their comprehension will be easier. Also, we present, in Section 2.4, a well known result which seems to be closely related to the other theorems, but whose proof is much easier.

2.1 The main step

In both proofs, a couple of (continuous, linear) integral operators S, T are constructed such that S + T is the identity, kS(χEf )k₂² ≤ α(ε)kfk₂², and kχFT fck2² ≤ β(ε)kfk2², where α(ε), β(ε)→ 0 as ε → 0. From such a situation, the Uncertainty Principle can be easily derived, as follows, using also Plancherel’s theorem and the trapezium rule.

kfk2²=k bfk2²=kχF^cfbk2²+kχFfbk2²

=kχF^cfbk2²+kχF( cSf + cT f )k2²

≤ kχF^cfbk2²+ 2kχFSfck2²+ 2kχFT fck2²

≤ kχF^cfbk2²+ 2kSfk2²+ 2β(ε)kfk2²

≤ kχF^cfbk2²+ 4kS(χE^cf )k2²+ 4kS(χEf )k2²+ 2β(ε)kfk2²

≤ kχF^cfbk2²+ 4kSk²kχE^cfk2²+ 4(α(ε) + β(ε))kfk2²

≤ max(1, 4kSk²)(k bfk_L²²_(F^c₎+kfk_L²²_(E^c₎) + 4(α(ε) + β(ε))kfk2²

Once this has been established, ε can be chosen small enough so that 4(α(ε)+

β(ε)) < 1. Then, subtracting 4(α(ε) + β(ε))kfk2² from both sides of the inequality and dividing by 1− 4(α(ε) + β(ε)) one gets the desired result.

2.2 A lemma related to Schur’s test

To obtain the conditionskS(χEf )k2² ≤ α(ε)kfk2²andkχFT fck2² ≤ β(ε)kfk2², one uses the following lemma.

2.1 Lemma. Let S, T be operators given by Sf (x) =

Z

A(x, y) f (y) dy and T f (y) =c Z

B(x, y) bf (y) dy , for measurable functions A, B : R^d× R^d→ C. Assume that A, B satisfy the following conditions, for some C > 0 and measurable sets E, F .

i. R |A(x, y)| dx < C for all y ∈ R^d, 6

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2.2. A lemma related to Schur’s test 7

ii. R |A(x, y)| dy < C for all x ∈ R^d, iii. R

E|A(x, y)| dy < Cε for all x ∈ R^d, iv. R |B(x, y)| dx < C for all y ∈ R^d, v. R |B(x, y)| dy < C for all x ∈ R^d, vi. R

F|B(x, y)| dx < Cε for all y ∈ R^d. It follows that S, T are bounded and

kS(χEf )k2 . ε^1/2kfk2, kχFT fck2. ε^1/2kfk2

hold.

Proof. First observe that, in general, for any operator K given by Kf (x) =

Z

k(x, y) f (y) dy ,

the conditionR |k(x, y)| dy ≤ C implies, by the Cauchy-Schwartz inequality,

|Kf(x)| ≤ Z

|k(x, y)| |f(y)| dy

= Z

|k(x, y)|^1/2|k(x, y)|^1/2|f(y)| dy

≤

Z

|k(x, y)| dy

¹₂ Z

|k(x, y)| |f(y)|²dy

¹₂

≤ C^1/2

Z

|k(x, y)| |f(y)|²dy

¹₂ .

So, in the particular case of S we have, using this observation (applied to property (iii )),

kS(χEf )k2² = Z

|S(χEf )|²

= Z

Z

E

A(x, y) f (y) dy

2

dx

≤ Z

Cε Z

E|A(x, y)| |f(y)|²dy dx .

An application of Fubini’s theorem and property (i ) shows this to be less than or equal to

Z

E

C²ε|f(y)|²dy = C²εkfk_L²²_(E)≤ C²εkfk2².

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8 2.3. A geometrical lemma

Similarly in the case of T . We apply our general observation to property (v ). Then use Fubini’s theorem and property (vi ), and finally apply Plancherel’s theorem, as follows.

kχFT fck2²= Z

F

| cT f|²

= Z

F

Z

B(x, y) bf (y) dy

2

dx

≤ Z

F

C Z

|B(x, y)| | bf (y)|²dy dx

≤ Z

C²ε| bf (y)|²dy

= C²εk bfk2² = C²εkfk2².

The proof of the boundedness of S, T is a bit easier and follows along the same lines (take, for example, E = R^d, ε = 1). It is known as Schur’s test (see [2, section 18.1]). Note that for this part (iii ) and (iv ) are not used.

2.3 A geometrical lemma

In this section, we present an easy lemma that will prove useful in later sections. But first, we need an even easier lemma about covering.

2.2 Lemma. Let A ⊂ R^d be open and let {xj} ⊂ A be a finite family of points such that B_j⁰ = B(x_j, r_j/3), for some r_j, cover A. Then, there is a subfamily{xji} of {xj} such that the balls Bji = B(xji, rji) cover A and the (smaller) balls B_j⁰_i = B(xji, rji/3) are disjoint.

Proof. (We follow [4, section 7.3].) Without loss of generality, we can suppose that the r_j’s are in decreasing order, i.e., r_j ≥ rj+1. We construct{ji} inductively. Put j₁ = 1, and for i > 1 let j_i be the smallest integer greater than ji−1 such that B_j⁰_i∩ (∪k<iB_j⁰

k) is empty. This process is finished in a finite number of steps. It is immediate that the balls B⁰_j

i are disjoint.

For each j there is some i such that j_i ≤ j and B_j⁰ intersects B_j⁰

i. Since the radius of Bji is three times larger than the radius of B_j⁰_i, B_j⁰ ⊂ Bji. Thus, A is covered by the balls B_j_i.

2.3 Lemma. Assume that ρ : R⁺ → R⁺ is continuous, that ε is positive, and that A⊂ R^d is ε-thin with respect to ρ, i.e. for all x∈ R^d

|A ∩ B(x, ρ(|x|))| ≤ ε|B(x, ρ(|x|))| . (2.3.1) Then there is a constant C_v > 0 such that, for all x ∈ R^d and for all r≥ ρ(|x|),

|A ∩ B(x, r)| ≤ Cvε|B(x, r)| , where C_v depends only on d.

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2.4. A well known result 9

Proof. (For this proof, we follow [3].) First, we prove that B(x, r) can be covered by balls B(y, ρ(|y|)/3) with ρ(|y|) ≤ 3r. If not, then there is some y∈ B(x, r) which is not covered and for which ρ(|y|) > 3r.

Let z = (1− t0)x + t₀y. Then z ∈ B(x, r) since both x and y belong to B(x, r), which is convex, and z is between them. Also, ρ(|z|) = 3r, so y is covered by B(z, ρ(|z|)/3); this contradicts the existence of y.

Thus, let {B(y, ρ(|y|)/3)} be a cover of B(x, r) with ρ(|y|) ≤ 3r. Since B(x, r) is compact, we can choose a finite subcover (thereby choosing finitely many points {xk}), and then apply Lemma 2.2 (p. 8). This allows us to choose a (finite) subcollection {xkl}l of the {xk}k so that the balls

B(x_k_l, ρ(|xkl|)/3)

are disjoint and B(x, r) is still covered by {B(xkl, ρ(|xkl|))}l. Note that, since ρ(|xk_l|) ≤ r, we also have that

∪lB(x_k_l, ρ(|xkl|)) ⊂ B(x, 2r) . Therefore,

X

l

|B(xkl, ρ(|xkl|))| = 3^dX

l

|B(xkl, ρ(|xkl|)/3)|

= 3^d| ∪lB(x_k_l, ρ(|xkl|)/3)|

≤ 3^d|B(x, 2r)|

= 6^d|B(x, r)| .

From this geometrical property, we can immediately derive the lemma, keeping in mind that A satisfies property (2.3.1):

|A ∩ B(x, r)| ≤X

l

|A ∩ B(xkl, ρ(|xkl|))|

≤ εX

l

|B(xkl, ρ(|xkl|)|

≤ Cvε|B(x, r)| , where C_v = 6^d depends only on d.

2.4 A well known result

We first examine the following well known version of the Uncertainty Prin- ciple, which seems to inspire the line of the proofs we will give later. The following statement is given as Lemma 4.2 in [5].

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10 2.4. A well known result

2.4 Theorem. Let ε > 0, and let E ⊂ R^d be ε-thin with respect to ρ≡ 1, i.e. |E ∩ B(p, 1)| < ε for all p ∈ R^d. Then, there exists some constant C > 0 such that, for all f ∈ L²(R^d) satisfying supp bf ⊂ B(0, 1), we have the estimate

kfkL²(E)≤ Cε^1/2kfk2.

2.5 Remarks. This is, of course, a version of the Uncertainty Principle, for it says that if the spectrum of f is concentrated in a compact set and ε is small, then f cannot be concentrated in the set E, and the situation gets worse as E gets smaller, i.e., as ε decreases.

Note, furthermore, that if ε is taken small enough so that Cε^1/2 < 1/√ 2, it follows that

Z

|f|² ≤ 1 2

Z

E^c

|f|²+ Z

B(0,1)^c

| bf|²

! .

(The last integral vanishes, obviously.) Put in this way, this result very closely resembles the ones we will discuss later.

Proof. Fix f ∈ L²(R^d) such that supp bf ⊂ B(0, 1). Let φ be a Schwartz function such that bφ = 1 on B(0, 1). Define the operator B : L²(R^d) → L²(E) by Bg = (g∗ φ)|E. Then, since supp ˆf ⊂ B(0, 1), we have f|E = ( ˆφ ˆf )ˇ|E = f ∗ φ|E = Bf .

Since B is given in integral form by Bg(x) = R χ_E(x)φ(x− y)g(y) dy, by Schur’s test [2, section 18.1] (see also Lemma 2.1 (p. 6) of the present paper), the norm of B is less than or equal to√

A₁A₂, where A₁= sup

x∈E

Z

|φ(x − y)| dy = kφk1 and A₂= sup

y∈R^d

Z

E

|φ(x − y)| dx .

We now prove that A2 . ε. Since φ is a Schwartz function, there is a constant C_φ such that |φ(x)| ≤ Cφ(1 +|x|)^−2d. Fix y ∈ R^d. Set Ω_k = E∩ B(y, 2^k)\B(y, 2^k⁻¹) for k > 0, and Ω₀ = E∩ B(y, 1). Then R^d =∪n≥0Ω_n. If we set ρ = 1, E satisfies the hypotheses of Lemma 2.3 (p. 8), so we can do the following estimations.

Z

E

|φ(x − y)| dx =

∞

X

k=0

Z

Ωn

|φ(x − y)| dx

≤

∞

X

k=0

Z

Ωn

Cφ(1 +|x − y|)^−2ddx

≤ Cφ|E ∩ B(y, 1)| +

∞

X

k=1

Cφ(1 + 2^k−1)^−2d|E ∩ B(y, 2^k)|

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2.4. A well known result 11

. Cφε +

∞

X

k=1

C_φ2^−2dkC_vε|B(y, 2^k)|

. Cφε +

∞

X

k=1

C_φ2^−dkCvε|B(y, 1)|

. ε Cφ(1 + Cv|B(y, 1)|).

Thus,kBk ≤√

A1A2 . ε^1/2, which is precisely the content of the theorem, for we have

kfkL²(E)=kBfkL²(E)≤ kBk kfk2. ε^1/2kfk2.

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3 The Shubin-Vakilian-Wolff theorem

We restate here the contents of the Shubin-Vakilian-Wolff theorem, which appeared as Theorem 2.1 in [5].

One first defines a set A⊂ R^d to be ε-thin if, for all x∈ R^d,

|A ∩ B(x, ρ(|x|))| ≤ ε|B(x, ρ(|x|))| , where ρ(t) = min(1, 1/t) for t > 0.

3.1 Theorem (Shubin, Vakilian, Wolff). There exists ε > 0 such that if E, F are ε-thin sets in R^d then there is a constant C > 0 such that, for any f ∈ L²(R^d),

Z

|f|² ≤ C

Z

E^c

|f|²+ Z

F^c

| bf|²

.

It is an open question whether the theorem holds for all ε < 1.

The reader may find it useful to look at the Formularium on page 37 while following the proof.

3.1 Construction of the operators

In view of what was done in Section 2.1, it is now necessary only to construct appropriate operators S, T .

Take ψ0: R^d→ R a radial, Schwartz function supported on B(0, 2) such that 0≤ ψ0≤ 1 everywhere and ψ0 = 1 on B(0, 1). For j > 0 define ψ_j(x) = ψ₀(x/2^j)− ψ0(x/2^j−1). Hence supp ψ_j ⊂ {x ∈ R^d: 2^j−1≤ |x| ≤ 2^j+1} and P

jψj = 1. Observe, moreover, thatkψjk1= 2^(j−1)dkψ1k1 for j > 1.

Let φ be the inverse Fourier transform of ψ₀; thus, φ ∈ S(R^d). Define φ_j(x) = 2^jdφ(2^jx), which yields (after a change of variable) kφjk1 =kφk1, and also bφj(x) = bφ(2^−jx), so that supp bφj ⊂ B(0, 2^j+1) and bφj = 1 on B(0, 2^j).

Define

Sf =

∞

X

j=0

ψ_j· (φj ∗ f),

T f =

∞

X

j=0

ψ_j· (f − φj∗ f) .

Since the supports of the ψ_j’s overlap at any given point in R^d for at most three different values of j, the sums are pointwise finite, and therefore pointwise convergent. Using Lemma 2.1 (p. 6), we will later prove that S and T are bounded. Remark also that, clearly, S + T is the identity operator.

12

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3.1. Construction of the operators 13

These operators can be represented as integrals, as follows. Explicitly, one has

Sf (x) =

∞

X

j=0

ψj(x) (φj ∗ f(x))

=

∞

X

j=0

ψj(x) Z

φj(x− y) f(y) dy, and also,

T f (x) =c





∞

X

j=0

ψj· (f − φj ∗ f)





V

(x)

=





∞

X

j=0

ψcj∗ ( bf− bφj· bf )



(x)

=

∞

X

j=0

Z

ψc_j(x− y) (1 − bφ_j(y)) bf (y) dy . Thus, if we define

A(x, y) =

∞

X

j=0

ψj(x)φj(x− y),

B(x, y) =

∞

X

j=0

cψj(x− y)(1 − bφj(y)),

we have that Sf (x) =R A(x, y) f (y) dy,T f (x) =c R B(x, y)f (y) dy.b

The following identity will save us much trouble in the rest of the proof.

B(x, y) =

∞

X

j=0

cψj(x− y)(1 − bφj(y))

=

∞

X

j=0

cψj(x− y)



 X

k>j

ψ_k(y)





=

∞

X

k=1

ψ_k(y)





k−1

X

j=0

cψj(x− y)





=

∞

X

k=1

ψ_k(y) φ_k−1(x− y) .

This has almost the same shape as A. Such a coincidence certainly accounts for the greatest elegance display in the proof of C. Shubin, R. Vakilian and T. Wolff.

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14 3.2. Boundedness properties of the operators

3.2 Boundedness properties of the operators

In this section, we provide proof for conditions (i ) through (vi ) in Lemma 2.1 (p. 6) for the functions A, B defined in the previous section. In view of the work done in Section 2.1, this will conclude the proof of Theorem 3.1 (p. 12).

To prove condition (i ), fix y ∈ R^d. We want to estimate R |A(x, y)| dx by some constant not depending on y. Let P be the set of j’s such that the distance between y and supp ψ_j is less than 1. Since there are not many numbers in P (actually, there cannot be more than three), we will content ourselves with a very coarse estimate for their corresponding contributions.

Since|ψj| ≤ 1 for all j, for j ∈ P we will use the estimate Z

|ψj(x) φj(x− y)| dx ≤ Z

|φj(x− y)| dx = kφk1.

For the rest of the j’s we want a better estimation, so we use the fact that φ is a Schwartz function. Hence, there is some constant C_φ such that, for all x ∈ R^d, |φ(x)| ≤ Cφ(1 +|x|)^−3d. This implies, by the definition of the φj’s, that |φj(x)| ≤ Cφ2^jd(1 + 2^j|x|)^−3d. Therefore, if j /∈ P , we can estimate

Z

|ψj(x) φ_j(x− y)|dx ≤ Z

|ψj(x)| Cφ2^jd(1 + 2^j|x − y|)^−3ddx

≤ Cφ2^jd2^−3jd Z

|ψj|

= C_φ2^jd2^−3jd2^(j−1)dkψ1k1

≤ Cφ2^−jdkψ1k1 ,

where we used that the distance between x ∈ supp ψj and y is greater or equal than 1, as well as the fact thatkψjk1 = 2^(j^−1)dkψ1k for j > 1.

Thus, Z

|A(x, y)| dx ≤



 X

j∈P

+X

j /∈P



 Z

|ψj(x)φ_j(x− y)| dx

≤ 3kφk1+ Cφkψ1k1

X

j≥0

2^−jd

which is constant, finite, and does not depend on y. Thus condition (i ) is proved.

Condition (ii ) is especially easy to prove in this case. Fix x∈ R^d. Then, there are at most three different values of j such that ψ_j(x)6= 0 and we get

Z

|A(x, y)| dy ≤

∞

X

j=0

Z

|ψj(x) φj(x− y)| dy ≤ 3kφk1 ,

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3.2. Boundedness properties of the operators 15

sincekφjk1=kφk1. This constant is, clearly, independent of x.

The proof of (iii ) is probably the most interesting part of the proof, since it is the only point at which we will use the condition of ε-thinness.

Nonetheless, it is easy.

Fix x∈ R^d. We want to estimate Z

E

|A(x, y)| dy ≤X

j≥0

Z

E

|ψj(x) φj(x− y)| dy

by ε times some constant not depending on x. There are at most three values of j such that ψj(x)6= 0. For these j’s, we will treat the summands R

E|ψj(x)φj(x− y)| dy separately; since the constant will be independent of x and j and since there are no more than three relevant j’s, we will be done.

So let, from now on, j be such that ψj(x)6= 0. Thus, 2^j−1 ≤ |x| ≤ 2^j+1 if j≥ 1, |x| ≤ 2 if j = 0.

We partion R^dinto Ω₀ = B(x, ρ(|x|)) and

Ω_n= B(x, 2ⁿρ(|x|))\B(x, 2ⁿ⁻¹ρ(|x|))

for n > 0. Then R^d=∪n≥0Ωn. Note that since E is ε-thin, by Lemma 2.3 (p. 8) there is a constant Cv > 0 (depending only on d) such that, for all n≥ 0,

|E ∩ B(x, 2ⁿρ(|x|))| ≤ Cvε|B(x, 2ⁿρ(|x|))| .

Furthermore, since φ is a Schwartz function, there is a constant C_φ⁰ > 0 such that, for all z∈ R^d,|φ(z)| ≤ C_φ⁰(1 +|z|)^−2d and, by the definition of φj, this implies|φj(z)| ≤ C_φ⁰ 2^jd(1+2^j|z|)^−2d. Moreover, since 2^j−1≤ |x| ≤ 2^j+1 for j > 0, we have 2^−(j+1) ≤ ρ(|x|) ≤ 2^−(j−1); the inequality remains true for j = 0. Thus, if we take for instance C⁰ ≥ 2^3dC_φ⁰, we have the estimate

|φj(z)| ≤ C⁰ρ(|x|)^−d(1 +|z|/ρ(|x|))^−2d.

Putting it all together, we get the following chain of inequalities.

Z

E

|ψj(x) φ_j(x− y)| dy ≤X

n≥0

Z

Ωn∩E|φj(x− y)| dy

≤X

n≥0

Z

Ωn∩E

C⁰ρ(|x|)^−d

1 +|x − y|

ρ(|x|)

_−2d dy

.X

n≥0

C⁰ρ(|x|)^−d2^−2nd|B(x, 2ⁿρ(|x|)) ∩ E|

.X

n≥0

C⁰ρ(|x|)^−d2^−2ndCvε|B(x, 2ⁿρ(|x|))|

.X

n≥0

C⁰ρ(|x|)^−d2^−2ndC_vε 2^ndρ(|x|)^d|B(x, 1)|

∼ ε CvC⁰|B(0, 1)|X

n≥0

2^−nd,

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16 3.2. Boundedness properties of the operators

which is ε times a finite constant independent of j and x. Thus, we are done.

As can be easily verified by the reader, properties (iv ) through (vi ) have almost exactly the same proofs, due to the identity

B(x, y) =

∞

X

k=1

ψk(y) φk−1(x− y) ,

discussed in the last section. This concludes the proof of the Shubin- Vakilian-Wolff theorem.

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4 Kovrizhkin’s generalization

In this section we discuss Kovrizhkin’s theorem. We state it in Section 4.1, and immediately afterwards proceed to discuss the properties of the definitions and of the theorem. The proof is then presented in Sections 4.3 through 4.7.

4.1 Definitions and statement

Let ε > 0. Two measurable sets E, F are said to be ε-thin with respect to ρ₁ and ρ₂ if they satisfy

|E ∩ D(x, ρ1(|x|))| ≤ ε|D(x, ρ1(|x|))|,

|F ∩ D(x, ρ2(|x|))| ≤ ε|D(x, ρ2(|x|))|

for all x∈ R^d.

Let ρ₁, ρ₂ : R⁺ → R⁺be two continuous, non-increasing functions. They are said to satisfy the Kovrizhkin condition if there exist positive constants C1, C2 such that

C₁ ρ₁

C2

ρ2(t)

≥ t for all t > 0. (4.1.1)

4.1 Theorem (Kovrizhkin). If ρ1 and ρ2 satisfy the Kovrizhkin condition, then there exists ε > 0 such that, for every pair of sets E, F that are ε- thin with respect to ρ₁ and ρ₂, there is a positive constant C that gives the estimate

Z

|f|² ≤ C

Z

E^c

|f|²+ Z

F^c

| bf|²

uniformly on f ∈ L²(R^d).

It is an open question whether the theorem is true for all ε < 1.

Examples of pairs of functions satisfying the hypotheses of the theorem are the functions ρ1(t) = min(1, t⁻ⁿ), ρ2(t) = min(1, t^−1/n), for integer n > 0.

Observe that the theorem is scale-invariant. Indeed, for fixed k > 0, if one lets ˜f (x) = f (kx), ˜E = kE, ˜F = ¹_kF , then ˜E, ˜F are ε-thin with respect to kρ₁(t/k) and ¹_kρ₂(kt), which still satisfy the Kovrizhkin condition. So the theorem holds.

Note also that letting ρ1(t) = ρ2(t) = min(1, t⁻¹), one gets the Shubin- Vakilian-Wolff theorem. One can even prove that Theorem 4.1 contains the Logvinenko-Sereda theorem as an extreme case. This is done in Section 4.8.

Kovrizhkin’s condition has a symmetry property. Let u = C2/ρ2(t);

then, it is a straightforward computation to show that equation (4.1.1) is 17

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18 4.2. Alternative characterization in terms of the Legendre transform

equivalent to

C₂ ρ₂

C1

ρ1(u)

≥ u. (4.1.2)

4.2 Remark. The constant C obtained in the theorem can be shown to be uniform on all E, F , once ρ₁, ρ₂ are fixed, if ε is taken small enough. Suppose that this is not true; then, there exist, for each n, a function fn ∈ L²(R^d) and sets En, Fnthat are εn-thin with respect to ρ1 and ρ2, with εn< ε/2ⁿ, such that

Z

|fn|² > n Z

E_n^c|fn|²+ Z

F_n^c|cf_n|²

! .

Let E =∪^∞n=1E_n and F =∪^∞n=1F_n. Then E and F are ε-thin with respect to ρ₁ and ρ₂. But we have

Z

|fn|²> n Z

E_n^c

|fn|²+ Z

F_n^c

|cf_n|²

!

≥ n

Z

E^c

|fn|²+ Z

F^c

|cf_n|²

,

for all n, thus contradicting the theorem.

4.2 Alternative characterization in terms of the Legendre transform

Now, in order to obtain another form of the hypotheses, we will use the Legendre transform, which is defined, for convex functions, as

Lf(t) = sup{tξ − f(ξ) : ξ > 0} (4.2.1) for f : R⁺→ R⁺. If f is convex and C², then f⁰⁰> 0 and for t∈ [f⁰(0),∞), the supremum (4.2.1) is attained (if at all) at the unique point ξ = (f⁰)⁻¹(t).

Differentiating the identityLf(t) = −f((f⁰)⁻¹(t)) + t(f⁰)⁻¹(t), we get (Lf)⁰ = (f⁰)⁻¹. (4.2.2) We will assume that ρ₁, ρ₂ are C¹, strictly decreasing and strictly positive. If we define ˜ρ₁(t) = ρ₁(C₂t)/C₁, then (4.1.1) becomes

1

˜ ρ₁

1 ρ2(t)

≥ t.

Equivalently,

1

˜ ρ1

₋₁

(t)≤ 1 ρ2(t).

Let g_i= 1/ρ_i and ˜g_i= 1/ ˜ρ_i, i = 1, 2; rewrite the inequality above as ( ˜g₁)⁻¹ ≤ g2.

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4.3. Outline of the proof 19

Denote by R

˜

g₁ some primitive of ˜g₁; since ˜g₁ is increasing, R

˜

g₁ is convex.

By (4.2.2), we can rewrite the equation above as (L

Z

˜

g1)⁰≤ g2.

Similarly, using the alternative form of Kovrizhkin’s condition, equation (4.1.2), one obtains,

(L Z

˜

g2)⁰≤ g1, where ˜g2(t) = C2/ρ2(C1t).

4.3 Outline of the proof

This proof of Theorem 4.1 (p. 17) is very similar to the one given for the Shubin-Vakilian-Wolff theorem in Section 3. The idea is the same one: The same partition of unity{ψj} is defined, and some functions φj closely related to it are constructed. Using these, the operators S, T are defined in the same way as before. Lemma 2.1 (p. 6) is then proved to be applicable, and the conclusion follows by what was shown in Section 2.1.

However, in this case there is no nice relation between the kernels of S and T , like the one discussed at the end of Section 3.1. This only makes things much less elegant and complicates the proofs of properties (iv ) through (vi ) in Lemma 2.1 (p. 6).

We make the constructions precise. Assume that we are given a pair of functions ρ1, ρ2 satisfying the Kovrizhkin condition, equation (4.1.1). ψj, j≥ 0, are defined in exactly the same way as before (see Section 3.1). The definition of the φ’s changes; in this case, we let φ = ˇψ₀ and, for j ≥ 0,

φ_j(x) =

C₁ ρ1(2^j−1)

d

· φ

C₁ ρ1(2^j−1)x

. We still have kφk1=kφjk1 for all j≥ 0.

For f ∈ L²(R^d), the operators S, T are also defined as in Section 3.1, and the same comments about convergence apply. Also, the kernels A, B have the same form.

The following three sections are devoted to the verification of the hypotheses required by Lemma 2.1 (p. 6). The reader may find it useful to look at the Formularium on page 37 while following the proof.

Although the identity discussed at the end of Section 3.1 does not hold for these operators, we have the following.

4.3 Lemma. The following identity holds:

B(x, y) =

∞

X

j=0

2^jdφ(2^j(x− y))

ψ0

ρ₁(2^j) C1

y

− ψ0

ρ₁(2^j−1) C1

y

.

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20 4.4. First easy verifications

Proof. Note that, since ψ₀is a radial function, φ is also radial and it is even, and thus, for all x∈ R^d, j ≥ 1,

cψj(x) = 2^jdcψ0(2^jx)− 2^(j−1)dψc0(2^j−1x)

= 2^jdφ(−2^jx)− 2^(j−1)dφ(−2^j−1x)

= 2^jdφ(2^jx)− 2^(j−1)dφ(2^j−1x).

Also, directly from the definition of φ_j, a change of variables implies bφ_j(x) = ψ₀(ρ₁(2^j⁻¹) x/C₁). So we have

B(x, y) =

∞

X

j=0

cψ_j(x− y) (1 − bφ_j(y))

= φ(x− y)(1 − bφ0) +

∞

X

j=1

2^jdφ(2^j(x− y)) − 2^(j^−1)dφ(2^j⁻¹(x− y))

(1− bφ_j(y))

=

∞

X

j=0

2^jdφ(2^j(x− y))(bφj+1(y)− bφj(y))

=

∞

X

j=0

2^jdφ(2^j(x− y))(ψ0(ρ1(2^j) y/C1)− ψ0(ρ1(2^j−1) y/C1)).

4.4 First easy verifications

In this section, we verify conditions (i ), (ii ), and (iv ) required by Lemma 2.1 (p. 6), which use techniques already developed in Section 3.2.

Regarding condition (i ), we observe that, once y ∈ R^d is fixed, there are at most four values of j such that the distance between supp ψj and y is less than, or equal to, 2^j−2; denote this set of j’s by P . Since φ is a Schwartz function, there is a constant C_φ such that|φ(x)| ≤ Cφ(1 +|x|)^−2d. Then, |φj(x)| ≤ CφC₁^dρ1(2^j−1)^−d(1 + C1ρ1(2^j−1)⁻¹|x|)^−2d and, if we let Bj = B(0, 2^j+1) (so that supp ψj ⊂ Bj),

Z

|A(x, y)| dx ≤



 X

j∈P

+X

j /∈P



 Z

|ψj(x) φj(x− y)| dx

≤ 4 kφk1+X

j /∈P

Z

Bj

|ψj(x)| Cφ

C₁^dρ₁(2^j⁻¹)^−d (1 +_ρ ^C¹

1(2^j⁻¹)|x − y|)^2ddx

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4.4. First easy verifications 21

≤ 4 kφk1+X

j /∈P

Z

Bj

C_φ C₁^dρ₁(2^j⁻¹)^−d (1 + _ρ ^C¹

1(2^j⁻¹)2^j−2)^2ddx

≤ 4 kφk1+

∞

X

j=0

C_φ C₁^dρ1(2^j−1)^−d (1 +_ρ ^C¹

1(2^j⁻¹)2^j⁻²)^2d 2^d(j+1)|B(0, 1)|,

since|Bj| = 2^d(j+1)|B(0, 1)|. To estimate the second term, we use the fact that ρ₁ is non-increasing. Let k ≥ −1 be the smallest integer such that C12^k≥ ρ1(2^k). Split the sum in two parts if k >−1 or just ignore the first part if k =−1. (We omit the constants Cφ and |B(0, 1)|.)





k

X

j=0

+

∞

X

j=k+1



2^d(j+1) C₁^dρ₁(2^j−1)^−d (1 +_ρ ^C¹

1(2^j−1)2^j−2)^2d . 2^kdC₁^dρ₁(2^k+1)^−d+

∞

X

j=k+1

2^−jdC₁^−dρ₁(2^j⁻¹)^d,

This sum is a finite constant not depending on y, so we are done.

For condition (ii ), for fixed x∈ R^d we have again at most three nonva- nishing ψj’s, leading to the estimate

Z

|A(x, y)| dy ≤ 3kφk1.

Now on to condition (iv ). Since ψ0 was taken to be a radial function, q(|x|) := ψ0(x) is well defined on R⁺. Fix y∈ R^d. Then, beginning with the identity in Lemma 4.3 (p. 19) and with an application of the fundamental theorem of calculus,

Z

|B(x, y)| dx

≤

∞

X

j=0

(ψ0(ρ1(2^j)y/C1)− ψ0(ρ1(2^j−1)y/C1)) Z

2^jd|φ(2^j(x− y))| dx

=kφk1

∞

X

j=0

(ψ0(ρ1(2^j)y/C1)− ψ0(ρ1(2^j−1)y/C1))

=kφk1

∞

X

j=0

Z ρ1(2^j⁻¹)|y|/C1

ρ1(2^j)|y|/C1

|q⁰(t)| dt

=kφk1

Z _∞

0

|q⁰(t)| dt

which is finite, since ψ0 is a Schwartz function, and independent of y, so condition (iv ) holds.

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22 4.5. The verification of conditions (iii) and (vi)

4.5 The verification of conditions (iii ) and (vi )

The verification of conditions (iii ) and (vi ) in Lemma 2.1 (p. 6) is again a very interesting part of the proof, since it is only here that the ε-thinness with respect to ρ1, ρ2 of the sets E, F will be used. Even more remarkably, the Kovrizhkin condition takes part in the proof only for the verification of condition (vi ). The general technique, however, is very similar to the one used in Section 3.2.

We now begin with the verification of condition (iii ). Let x∈ R^d. There are, again, at most three values of j such that ψ_j(x) does not vanish, so it will suffice to prove that R

E|φj(x− y)| dy ≤ Cε for some constant C that does not depend on x, when x∈ supp ψj, i.e., 2^j⁻¹≤ |x| ≤ 2^j+1.

We do some preparations. Let N ≥ 0 be the smallest integer such that 2^N ≥ C1, and let Cφ> 0 be such that, for all x∈ R^d,|φ(x)| ≤ Cφ(1+|x|)^−2d, so that one also gets

|φj(x− y)| ≤ Cφ

C1

ρ₁(2^j⁻¹)

d

1 + C1

ρ₁(2^j⁻¹)|x − y|

−2d

. Let Ω0= E∩ B(x, 2^Nρ1(|x|)/C1) and, for n≥ 1,

Ω_n= E∩ B(x, 2ⁿρ₁(|x|)/C1)\B(x, 2ⁿ⁻¹ρ₁(|x|)/C1).

Finally, let C_v be the constant given by Lemma 2.3 (p. 8). Now, we have the following chain of inequalities.

Z

E|φj(x− y)| dy

≤ Cφ

Z

E

C₁^dρ1(2^j−1)^−d (1 +_ρ^C¹^|x−y|

1(2^j−1))^2ddy

= Cφ

Z

ΩN

+

∞

X

n=N +1

Z

Ωn

!

C₁^dρ1(2^j−1)^−d (1 + _ρ^C¹^|x−y|

1(2^j−1))^2ddy . CφC₁^dρ₁(2^j−1)^−d|E ∩ B(x, 2^Nρ₁(|x|)/C1)|

+ Cφ

∞

X

n=N +1

C₁^dρ₁(2^j−1)^−d (1 + ^C¹⁽²ⁿ⁻¹_ρ ^ρ¹^(|x|)/C¹⁾

1(2^j−1) )^2d|E ∩ B(x, 2ⁿρ1(|x|)/C1)| . Cφ

C1

ρ₁(2^j⁻¹)

d

|B(x, 1)| 2^Nρ1(|x|) C₁

d

Cvε

!

+ C_φ

∞

X

n=N +1

C₁^dρ1(2^j−1)^−d (1 + ²ⁿ_ρ⁻¹^ρ¹⁽^|x|)

1(2^j⁻¹) )^2d |B(x, 1)| 2ⁿρ1(|x|) C₁

d

Cvε

!

Since ρ1(|x|) · ρ1(2^j−1)⁻¹ ≤ 1, the first term is easily seen to be smaller than a constant independent of x, j, times ε. We proceed to show that the sum in

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4.5. The verification of conditions (iii) and (vi) 23

the second term is finite; if we let a = ρ₁(|x|)/ρ1(2^j⁻¹), then we can rewrite the sum as a constant times

ε

∞

X

n=N +1

a^d2^(n−1)d

(1 + 2ⁿ⁻¹a)^2d. (4.5.1) The sum is estimated by the integral

Z _∞

1

a^d2^td

(1 + 2^ta)^2d dt . Z _∞

1

a^dy^d (1 + ya)^2d

dy y . We changed variables y = 2^t. Now, since a≤ 1, we write

Z _∞

1

= Z 1/a

1

+ Z _∞

1/a

where that the first integral is calculated to be Z 1/a

1

a^dy^d⁻¹

(1 + ay)^2ddy≤ Z 1/a

1

a^d(1/a)^d⁻¹

1 + ay dy = a

log 2

1 + a

.

This is clearly bounded uniformly on a. The second integral is Z _∞

1/a

a^dy^d−1

(1 + ay)^2ddy . Z _∞

1/a

1

a^dy^d+1dy = 1 d.

Thus, we are done, since these estimates do not depend on x or j.

The proof for condition (vi ) is similar. Fix y ∈ R^d. In view of the identity proved in Lemma 4.3 (p. 19), since there are at most three different values of j such that

ψ0(ρ1(2^j)y/C1)− ψ0(ρ1(2^j−1)y/C1)6= 0 , (4.5.2) it will suffice to prove thatR

F2^jd|φ(2^j(x− y))| ≤ C ε for some constant C independent of y —for y satisfying equation (4.5.2)—.

If y ∈ supp ψ0(ρ₁(2^j)y/C₁)− ψ0(ρ₁(2^j⁻¹)y/C₁), then C₁/ρ₁(2^j⁻¹) ≤

|y| ≤ 2 C1/ρ₁(2^j). The Kovrizhkin condition, equation (4.1.1), then gives

ρ2(|y|) ≤ ρ2

C1

ρ₁(2^j⁻¹)

≤ C2

2^−(j−1). Thus, if now Ω⁰₀= F ∩ B(y, ρ2(|y|)) and

Ω⁰_n= F ∩ B(y, 2ⁿρ₂(|y|))\B(y, 2ⁿ⁻¹ρ₂(|y|)) ,

The uncertainty principle for functions with sparse support and spectrum

The uncertainty principle for functions with sparse support and spectrum

Rodolfo A. Ríos Zertuche R. Z.

U.U.D.M. Project Report 2005:7

Department of Mathematics

Uppsala University

The Uncertainty Principle for functions with sparse support and spectrum

Contents

1 Introductory remarks

2 Some preparations

3 The Shubin-Vakilian-Wolff theorem

4 Kovrizhkin’s generalization