SUPPORTS OF REPRESENTATIONS IN THE COHEN CLASS 1

SUPPORTS OF REPRESENTATIONS IN THE COHEN CLASS ´ PAOLO BOGGIATTO, CARMEN FERNANDEZ AND ANTONIO GALBIS

Abstract. In this paper we consider a version of the uncertainty principle concerning limitations on the supports of time-frequency representations in the Cohen class. In particular we obtain various classes of kernels with the property that the corresponding representations of non trivial signals cannot be compactly supported. As an application of our results we show that a linear partial differential operator applied to the Wigner distribution of a function f 6= 0 in the Schwartz class cannot produce a compactly supported function.

1. Introduction In this paper we consider a version of the uncertainty principle concerning limitations on the supports of time-frequency representations in the Cohen class. The time-frequency representations considered in this paper are quadratic in the sense that they are defined through a sesquilinear map as follows. We first consider a map T : S(Rd ) × S(Rd ) → S 0 (R2d ) acting on pairs (f, g) of signals (i.e. functions) in S(Rd ), which is linear in f and conjugate linear in g. The tempered distribution T (f, g) can be interpreted as a linear time-frequency representation of the signal f associated to a window g. Then Q(f ) := T (f, f ), is quadratic in f and does not depend on an auxiliary window g. This is the way followed by Wigner, who in 1932 introduced what now is called Wigner distribution in the setting of quantum mechanics. In 1948 the use of the Wigner distribution was extended to signal theory by Ville. In 1933 Kirkwood [29] came up with another quadratic distribution (rediscovered by Rihaczek in 1968) arguing that it was simpler to use than the Wigner distribution. Many alternatives were introduced, like the one of Margeneau and Hille, Page, Choi and Williams, among others. In 1966 Cohen (see for instance [9]) gave a unified approach, which can be formulated in a simple manner, and proposed a general method for obtaining a family of joint time-frequency distributions. Quadratic or sesquilinear time-frequency representations gained some popularity in engineering after a series of papers by Claasen and Mecklenbräuker 2000 Mathematics Subject Classification. Primary 47G30. Secondary 35S05, 42B10, 44A35, 47B38. Key words and phrases. Time-frequency representations; Cohen class; Wigner distribution; uncertainty principle. 1

2

´ PAOLO BOGGIATTO, CARMEN FERNANDEZ AND ANTONIO GALBIS

[6, 7, 8]. Following Gröchenig [19], a quadratic time-frequency representation Q belongs to the Cohen’s class if it is of the form Q(f ) = σ ∗ W (f ) for some tempered distribution σ ∈ S 0 (R2d ) called the kernel of Q, where W (f ) denotes the Wigner distribution of the function f ∈ S(Rd ). We refer to Section 2 for the definition of the convolution. This representation is covariant with respect to time-frequency shifts of the signal. The relevance of the Cohen class is due to the fact that every quadratic covariant time-frequency representation, satisfying some weak continuity assumption, belongs to this class [19, Theorem 4.5.1]. In particular, the action of a linear partial differential operator with constant coefficients on W (f ) can be written in the above way with a kernel that is supported at {0}. We refer to [21, 6, 7, 8, 32], aimed at engineering audience, and also to [26, 27]. Some properties of the representation Q are reflected as simple constraints on the kernel σ. For a systematic study of this issue we refer to [8, 9, 15, 25, 27]. From the 1970’s onwards the study of uncertainty principles for timefrequency representations has received the attention of several authors (see for instance [3, 4, 13, 14, 23, 28] for some recent papers). Any time-frequency representation satisfies some appropriate version of the uncertainty principle. However, although it is well known that the Wigner distribution of a signal f ∈ L2 (Rd ) cannot be supported on a compact set (even on a set with finite Lebesgue measure [23, 28, 36]), as far as we know there is no general result of this type for arbitrary representations in the Cohen class. In fact, very little is known about uncertainty principles for the Cohen class [30]. The purpose of this paper is to propose some results in this direction. Our main interest is to investigate under which conditions the support of a representation in the Cohen class of a non-trivial signal cannot be compact. In some situations this conclusion is not hard to reach. For instance, let us assume that the representation Q satisfies the marginal property ([9, 30]), that is, for every f ∈ S(Rd ) we have ∫ ∫ 2 Q(f )(x, ω)dω = |f (x)| , Q(f )(x, ω)dx = |fb(ω)|2 Rd

Rd

for every x, ω ∈ Rd , where fb denotes the Fourier transformation of f. If Q(f ) is supported on a compact set K = K1 × K2 then the previous identities imply that f is supported on K1 and fb is supported on K2 , which forces the signal f to be identically null. In Proposition 9 we will show that the support of Q(f ) cannot be compact if the representation satisfies Moyal’s formula. As a third example we could consider the action of the Cauchy-Riemann operator on the Wigner distribution of a signal f ∈ L2 (R). More precisely, we consider Qσ : L2 (R) →

SUPPORTS OF REPRESENTATIONS IN THE COHEN CLASS

3

S 0 (R2 ) defined by Qσ (f ) = ∂W (f ). Then the restriction of Qσ to S(R) is a time-frequency distribution in the Cohen class as defined above with a kernel σ that is supported at {0}. We recall that the classical Weyl’s Lemma asserts that the distributional solutions of the CauchyRiemann equations are actually holomorphic functions (see for instance [12, Theorem 4.7] or page 216 in [34]). Hence Qσ (f ) = ∂W (f ) being compactly supported for f ∈ L2 (R) would mean that W (f ) coincides almost everywhere with a holomorphic function outside a closed disc. As W (f ) is real-valued and it is holomorphic outside a disc it is constant outside this disc. Further W (f ) ∈ L2 (R2 ) [19, 4.3.2] thus W (f ) is supported in a disc. As the Wigner distribution satisfies the marginal property, we conclude that f = 0. A more difficult example arises when we ask whether the Laplacian of the Wigner distribution of a signal f ∈ S(R) can be compactly supported or not. We answer this question in Theorem 22. Let us observe that the study of the support of P (D)W (f ), where P (D) is a linear partial differential operator, gives information about the regularity properties of W (f ). The outline of the paper is following. After a brief review of the basic notions that we shall need, we present our results in Sections 2 and 3, which essentially consist of a variety of cases where, under suitable hypothesis, the representation Qσ (f ) of a non-trivial signal f cannot have compact support. More precisely, in Section 2 we show that this is the case under some specific conditions on the supports of the signal f and the Fourier transform of the kernel σ as well as for f in suitable Gelfand-Shilov spaces. In Section 3 we focus on the role of compactness of the support of kernels and/or signals for obtaining the desired results, emphasizing the case where the kernel σ ∈ E 0 (R2 ) is a compactly supported distribution with the property that the corresponding convolution operator D0 (R2 ) → D0 (R2 ), u 7→ σ ∗ u, is hypoelliptic. As an application we show then that a linear partial differential operator applied to the Wigner distribution of a function f ∈ S(R) \ {0} cannot produce a compactly supported function. For the background about time-frequency representations we refer to Gröchenig [19]. The references for convolution operators and related topics are [22, 33, 34].

2. Preliminary results for kernels in S 0 (R2d ) We start with some notations and definitions which are used in the sequel. Our notation for spaces of test functions and distributions is the one in [33, 34]. For instance, D(K) is the space of C ∞ functions on Rd supported on the compact set K, D(Rd ) is the space of all compactly supported C ∞ functions on Rd , D0 (Rd ) is the space of distributions.

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The support of a distribution µ ∈ D0 (Rd ) is the complementary of the largest open set Ω such that (µ, ϕ) = 0 for each ϕ ∈ D(Rd ) with support contained in Ω. The support of a locally integrable function h will be understood as the support of the distribution associated to h. If h is continuous this coincides with its support as a function. E 0 (Rd ) denotes the space of all compactly supported distributions and it is the dual space of E(Rd ) = C ∞ (Rd ). We denote by S(Rd ) the Schwartz space of all C ∞ and rapidly decreasing functions and by S 0 (Rd ) its dual space. As usual, δ denotes the Dirac’s delta function. We use parenthesis ( , ) to denote the bilinear form defined by the dual pairs (D(Rd ), D0 (Rd ), (C ∞ (Rd ), E 0 (Rd )) and (S(Rd ), S 0 (Rd )) whereas h , i means the extension of the inner product of L2 (Rd ) to these dual pairs, that is hf, gi = (f, g). We refer to [33, 34] for the properties of the convolution defined next. Definition 1. Let us assume that (a) σ ∈ S 0 (Rd ) and f ∈ S(Rd ) or (b) σ ∈ D0 (Rd ) and f ∈ D(Rd ). The convolution σ ∗ f is the C ∞ function defined as σ ∗ f (x) := (σ, Tx f˜), x ∈ Rd , where Tx f (t) = f (t − x) and f˜(t) = f (−t). The convolution of a compactly supported distribution µ ∈ E 0 (Rd ) and a distribution σ ∈ D0 (Rd ) is the distribution µ ∗ σ ∈ D0 (Rd ) defined by (µ ∗ σ, ϕ) = (µ, σ ˜ ∗ ϕ), ϕ ∈ D(Rd ) where (˜ σ , ϕ) = (σ, ϕ). ˜ Definition 2. The convolution operator Tµ : D0 (Rd ) → D0 (Rd ), Tµ (σ) = µ ∗ σ, is said to be hypoelliptic if the condition Tµ (σ) ∈ C ∞ (Rd ) implies σ ∈ C ∞ (Rd ). ∑ α In the case that µ = |α|≤m aα ∂ δ is a finite linear combination of derivatives of the Dirac delta function then Tµ coincides with the lin∑ ear partial differencial operator |α|≤m aα ∂ α . According to [22], this operator is hypoelliptic if, and only if, its kernel consists of C ∞ functions. Definition 3. The cross-Wigner distribution of f, g ∈ L2 (Rd ) is ∫ t t W (f, g)(x, ω) = f (x + )g(x − )e−2πiωt dt, x, ω ∈ Rd . 2 2 Rd When f = g we write W (f ) = W (f, f ).


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It happens that W (f, g) ∈ L2 (R2d ) and it is uniformly continuous [19, 4.3.2]. The cross-Wigner distribution maps S(Rd )×S(Rd ) into S(R2d ). We recall that our definition of the Fourier transform for f ∈ S(Rd ) is ∫ b f (ω) = f (t)e−2πiωt dt, ω ∈ Rd . Rd

It defines an isomorphism F : S(Rd ) → S(Rd ), f 7→ fb, which extends to an isomorphism F : S 0 (Rd ) → S 0 (Rd ). We still denote by σ b = F(σ) the Fourier transform of the tempered distribution σ. The cross-Wigner distribution can be extended as a continuous map from S 0 (Rd ) × S 0 (Rd ) into S 0 (R2d ) as follows [19, 4.3.3]

hW (σ, µ), F i := σ ⊗ µ, Ts−1 F2−1 F , F ∈ S(R2d ), where F2 denotes the partial Fourier transform ∫ F2 F (x, ω) = F (x, t)e−2πiωt dt, x, ω ∈ Rd Rd

and Ts is the symmetric coordinate change defined by t t Ts F (x, t) = F (x + , x − ), x, t ∈ Rd . 2 2 This extension satisfies Moyal’s formula, that is hW (σ, µ), W (f, g)i = hσ, f ihµ, gi, where σ, µ ∈ S 0 (Rd ), f, g ∈ S(Rd ). Definition 4. Given σ ∈ S 0 (R2d ), the quadratic time-frequency representation Qσ is defined as Qσ : S(Rd ) → S 0 (R2d ), Qσ (f ) = σ ∗ W (f ), f ∈ S(Rd ).

(1)

We say that Qσ is a representation in the Cohen class and we call σ the Cohen kernel of Qσ . In fact, Qσ (f ) is a C ∞ function for every f ∈ S(Rd ). The Fourier transform of the cross-Wigner distribution of two funtions f, g ∈ L2 (Rd ) can be expressed in terms of the cross-ambiguity function (see for instance [19, 4.3.4]) as ∫ ν ν \ W (f, g)(ξ, ν) = A(f, g)(−ν, ξ) = f (t − )g(t + )e−2πitξ dt 2 2 Rd ∫ =

ξ ξ fb(ω + )b g (ω − )e−2πiων dω 2 2 Rd (2)

for ξ, ν ∈ Rd . Let us assume that σ b ∈ L∞ (R2d ), that∫ is, there is g ∈ L∞ (R2d ) ⊂ S 0 (R2d ) with the property that hb σ , f i = R2d g(x)f (x) dx for every f ∈

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S(R2d ). Then Qσ : S(Rd ) → S 0 (R2d ) can be extended to a continuous map Qσ : L2 (Rd ) → L2 (R2d ) by ( ) \ Qσ (f ) = F −1 σ b·W (f ) , f ∈ L2 (Rd ). We say that Qσ satisfies Moyal’s formula if σ b ∈ L∞ (R2d ) and hQσ (f ), Qσ (g)i = | hf, gi |2 for every f, g ∈ L2 (Rd ). It is well-known that W (f ) cannot be compactly supported even in the case that f belongs to S 0 (Rd ) \ {0} (see [14]). The situation is completely different for representations in Cohen class, as the following example shows. Example 5. There are σ ∈ S 0 (R2d ) \ {0} and f ∈ S(Rd ) \ {0} such that Qσ (f ) = 0. In fact, for every f ∈ S(Rd ) and ξ, ν ∈ Rd , ∫ ν ν \ W (f )(ξ, ν) = f (t − )f (t + )e−2πiξt dt. 2 2 Rd \ Hence, taking f supported by [−a, a]d we have that W (f ) is supported d d 0 2d by R × [−2a, 2a] . If we consider σ b ∈ S (R ) supported outside the strip Rd × [−2a, 2a]d , then \ Qσ (f ) = F −1 (b σW (f )) = 0. Thus, it is natural to ask whether it is possible to find a tempered distribution σ and a function f in the Schwartz class S(Rd ) such that Qσ (f ) has non-trivial compact support. We denote by Π1 the projection on the ξ−axis defined by Π1 : Rd × Rd → Rd , (ξ, ν) 7→ ξ. Similarly, Π2 is the projection on the ν−axis defined by Π2 : Rd × Rd → Rd , (ξ, ν) 7→ ν. Theorem 6. Suppose that σ ∈ S 0 (R2d ) and f ∈ S(Rd ) are such that supp σ b 6= R2d or (supp f − supp f ) 6= Rd . Then the support of Qσ (f ) is either empty or non-compact. Proof. Assume that Qσ (f ) is compactly supported. By the Paley\ Wiener-Schwartz Theorem [22, 7.3.1] W (f ) · σ b, which makes sense as \ multiplication between the Schwarz function W (f ) ∈ S(R2d ) and the tempered distribution σ b ∈ S 0 (R2d ), is an analytic function. We have \ then two possibilities: either W (f ) · σ b = 0 identically or its support is


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\ the whole R2d . In case supp σ b 6= R2d we conclude that W (f ) · σ b = 0, 2d \ hence Qσ (f ) = 0. Assume now that supp σ b = R . If W (f )· σ b = 0, then \ W (f ) = 0 hence f ≡ 0. If not we necessarily have supp W (f ) = R2d , hence from (2), as f and W (f ) are continuous functions, we have \ Rd = Π2 supp W (f ) ⊂ (supp f − supp f ).

This is a contradiction.

Remark 7. By [19, 4.3.2], W (fb)(x, ω) = W (f )(−ω, x). Therefore, given σ ∈ S 0 (R2d ) and f ∈ S(Rd ), it turns out that Qσ (fb)(x, ω) = QUσ (f )(−ω, x) where (Uσ, F ) = (˜ σ , UF ) and UF (s, t) = F (t, −s) is the action of the rotation operator U on the function F on R2d . Hence in the previous theorem the condition on f can be replaced by the same condition on fb. Observe that Theorem 6 also holds for functions f ∈ L2 (Rd ) under the additional assumption that σ b ∈ L∞ (R2d ). The proof in this case is more cumbersome. In the special case that σ b is compactly supported the conclusion of Theorem 6 is more simply obtained from the fact that \ Qσ (f ) = F −1 (W (f ) σ b) \ and W (f ) σ b is a compactly supported distribution. Hence, Qσ (f ) is the restriction of an entire function on C2d and therefore cannot have proper compact support. We recall that the short-time Fourier transform of a function f ∈ L2 (Rd ) with respect to a window g ∈ L2 (Rd ) \ {0} is ∫ Vg f (x, ω) = f (t)g(t − x)e−2iπω dt, x, ω ∈ Rd . (3) Rd

A good reference for its basic properties and its relation with the crossWigner distribution is [19]. Example 8. Let ϕ, ψ ∈ S(Rd ). The generalized spectrogram is defined in [2] as the sesquilinear form Specϕ,ψ (f, g) := Vϕ f · Vψ g, f, g ∈ S(Rd ). Then Q : S(Rd ) → S 0 (Rd ) defined by Q(f ) = Specϕ,ψ (f, f ) is a quadratic representation in the Cohen class. If ϕ, ψ ∈ S(Rd ) \ {0} are such that supp ϕ b − supp ψb 6= Rd or supp ψ − supp ϕ 6= Rd , (4) then the support of Q(f ) is either trivial or non-compact.

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˜ Using the expression In fact, according to [2], Q = Qσ for σ = W (ϕ, ˜ ψ). of the Fourier transform of the kernel σ as in (2) it is easy to check that the hypothesis imply supp σ b 6= R2d . Theorem 6 gives the conclusion. Proposition 9. Let σ ∈ S 0 (R2d ) be such that Qσ : L2 (Rd ) → L2 (R2d ) satisfies Moyal’s relation. Then, for every f ∈ L2 (Rd ) \ {0}, Qσ (f ) cannot be compactly supported. Proof. First we note that for each x, ω ∈ Rd , ∫ Qσ (f )(s, t)Qσ (f )(s − x, t − ω)d(s, t) R2d ∫ = Qσ (f )(s, t)Qσ (Mω Tx f )(s, t)d(s, t) ∫R2d 2 f (t)Mω Tx f (t)dt = |Vf f (x, ω)|2 . = Rd

Therefore if Qσ (f ) has compact support we get that Vf f (x, ω) has also compact support, which implies f = 0 ([28, 36]). To close this section we analyze the support of representations in the Cohen class when functions are taken is some subspaces of S(Rd ) invariant under Fourier transform and whose elements are analytic functions. Hence, all non-zero functions in these classes satisfy supp f = supp fb = Rd . Definition 10. (Gelfand-Shilov spaces) Let α, β ≥ 0 be real numbers, the Gelfand-Shilov space Sαβ (Rd ) is defined as the set of all C ∞ functions on Rd for which there exist A, B ∈ Rd+ and C > 0 such that ∂µ f (x)kL∞ (Rd ) ≤ CAλ B µ λαλ µβµ ∂xµ for every multi-indices λ ∈ Nd0 , µ ∈ Nd0 . kxλ

The space Sαβ (Rd ) is non trivial if and only if α + β > 1 or if α + β = 1 and α, β 6= 0. In this case, Sαβ (Rd ) is dense in the Schwartz space S(Rd ) and the Fourier transform defines an isomorphism F : Sαβ (Rd ) −→ Sβα (Rd ). 1/2

For 1/2 ≤ α < 1, functions in Sαα (Rd ) are real analytic. S1/2 (Rd ) equals the space considered by Bruijn [5]. Moreover Gelfand-Shilov spaces Sαα (Rd ) are invariant under the Wigner transform, that is if f, g ∈ Sαα (Rd ) and α ≥ 1/2 then W (f, g) ∈ Sαα (R2d ). For a more complete treatment we refer e.g. to [18] or [35]. Proposition 11. Given f ∈ Sαα (Rd ), 1/2 ≤ α < 1, and σ ∈ S 0 (R2d ) such that supp σ b has non empty interior, then if supp Qσ f 6= R2d we have f ≡ 0.


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Proof. We already know that W (f ) ∈ Sαα (R2d ) from where it follows that the convolution Qσ (f ) = σ ∗ W (f ) is real analytic. As \ supp Qσ (f ) 6= R2d we have Qσ (f ) = 0. Thus σ b·W (f ) = 0 which \ implies W (f ) is identically zero on a set with non-empty interior. On \ the other hand, W (f ) ∈ Sαα (R2d ), hence it is a real analytic function. \ We conclude that W (f ) = 0 and f = 0. 3. Compactly supported Cohen kernels We present in this section some results where the compactness of the support of the Cohen kernel σ and/or the function f play, in different ways, a crucial role to establish limitations on the support of Qσ (f ). Proposition 12. Let σ ∈ S 0 (R2d ) and f ∈ S(Rd ) and assume that they are both compactly supported. If Qσ (f ) is supported on a set of finite Lebesgue measure, then either f ≡ 0 or σ ≡ 0. Proof. For sake of simplicity, we write the proof in dimension 1. The general case follows the same line. We first show that Qσ (f )(x, ·) is real analytic for every x ∈ R. To this end, observe first that Qσ (f )(x, ·) is a C ∞ function. It is thus enough to show that for each x ∈ Rd there is b > 0 such that for every ω ∈ R, j ∂ ≤ Cbj Q (f )(x, ω) σ ∂j ω so that Taylor’s formula gives the analyticity. First, as σ is compactly supported, there exist C, R > 0 and an integer N ≥ 0 such that, for every ϕ ∈ C ∞ k+l ∂ |(σ, ϕ)| ≤ C sup sup k l ϕ(u, v) . k,l≤N ∂ u∂ v |u|,|v|≤R

But

j ( ) ∂ ( ∂j ) Q (f )(x, ω) = σ , W (f ) (x − u, ω − v) σ u,v ∂j ω ∂j ω so that we need to estimate ∂ j+k+l ∂ j ω∂ k u∂ l v W (f )(x − u, ω − v) for k, l ≤ N and |u|, |v| ≤ R. Now, there is an a such that f is C ∞ with support in [−a, a]. Further, there is a constant C(N ) such that all derivatives of f up to order N are bounded by C(N ). Moreover, ∫ ( t) t) ( W (f )(x − u, ω − v) = f x − u + f x − u − e−2iπωt e2iπvt dt 2 2 R

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from which one deduces that ∂ j+k+l W (f )(x ∂ j ω∂ k u∂ l v

− u, ω − v)

∫ j

=

j+l

(−1) (2iπt) R

[ ] ( ∂k t) ( t ) −2iπωt 2iπvt f x−u+ f x−u− e e dt. ∂ku 2 2

Note that, as f is supported in [−a, a] we only integrate over t ∈ [−2(a + |u| + |x|), 2(a + |u| + |x|)]. Further, using Leibnitz’ Formula, we get k ∂ [ ( t) ( t )] N 2 ∂ k u f x − u + 2 f x − u − 2 ≤ 2 C(N ) for k ≤ N. From this, we deduce that, for k, l ≤ N and |u| ≤ R, ∫ ∂ j+k+l N 2 |2πt|j+N dt ∂ j ω∂ k u∂ l v W (f )(x − u, ω − v) ≤ 2 C(N ) |t|≤2(a+R+|x|)

=

2N +1 C(N )2 ( 4π(a N +j+1

)j+N +1 + R + |x|) .

Therefore Qσ (f )(x, ·) is real analytic. On the other hand, for almost every x ∈ R the set {ω ∈ R : ω ∈ supp Qσ (f )(x, ·)} has finite Lebesgue measure. Hence Qσ (f )(x, ·) = 0 for almost every x ∈ R, which implies that Qσ (f ) = 0 and the conclusion follows. We now restrict our attention to the case d = 1. Let σ ∈ S 0 (R2 ) be given and assume that there is a compactly supported distribution µ such that σ ∗µ = δ i.e. σ is a fundamental solution of a convolution operator Tµ : D0 (R2 ) → D0 (R2 ), Tµ (ν) = ν ∗ µ, with µ compactly supported. Then if Qσ (f ) is compactly supported, so is W (f ) = µ ∗ Qσ (f ) thus f = 0. We note that when σ has compact support we cannot find a compactly supported distribution µ such that σ ∗ µ = δ except that σ is a translation of the Dirac delta function. However, under the additional hypothesis that the convolution operator Tσ is hypoelliptic, we will be able to obtain the non compactness of the support of Qσ (f ) using the existence of compactly supported parametrices (see [22]). Lemma 13. Let σ ∈ E 0 (R2 ) be a compactly supported distribution such that the convolution operator Tσ : D0 (R2 ) → D0 (R2 ), Tσ (µ) = µ ∗ σ, is hypoelliptic. We suppose that f ∈ S(R) and Qσ (f ) is compactly supported. Then, there are test functions ψ1 , ψ2 , λ1 , λ2 ∈ D(R) such that |f |2 = λ1 + |f |2 ∗ ψ1 and |fb |2 = λ2 + |fb |2 ∗ ψ2 .


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Proof. From the hypoellipticity of Tσ we deduce that there is E ∈ D0 (R2 ), which is a C ∞ function outside [−a, a]2 for some a > 0, such that E ∗ σ = δ ([22, 16.6.5]). Now we fix Ψ ∈ D(R2 ) equals one on a neighborhood of [−a, a]2 . Then (ΨE) ∗ σ = δ + (Ψ − 1)E ∗ σ. Since (Ψ − 1)E is a C ∞ function then we finally obtain ϕ := (1 − Ψ)E ∗ σ ∈ D(R2 )

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and (δ − ϕ) ∗ W (f ) = (ΨE) ∗ σ ∗ W (f ) = (ΨE) ∗ Qσ (f ) is a compactly supported C ∞ function. Let A > 0 be given so that (δ − ϕ) ∗ W (f ) vanishes outside [−A, A]2 and fix |x| > A. Then ∫ 2 |f (x)| = W (f )(x, ξ) dξ R

∫ =

(

R

∫ =

(

R

∫ =

R2

∫ =

R2

) W (f ) ∗ ϕ (x, ξ) dξ ∫ R2

) W (f )(x − s, ξ − t)ϕ(s, t) dsdt dξ

( ϕ(s, t)

∫ R

W (f )(x − s, ξ − t) dξ

)

dsdt

( ) ϕ(s, t)|f |2 (x − s) dtds = |f |2 ∗ ψ1 (x)

where

∫ ψ1 (s) =

ϕ(s, t) dt. R

Then λ1 := |f |2 − |f |2 ∗ ψ1 is compactly supported. The argument for |fb |2 is similar. Lemma 14. Let h ∈ S(R) and ψ, λ ∈ D(R) be such that h = λ + h ∗ ψ. Then there are b > 0, C > 0 such that (i) b h can be analytically extended to the strip |Imz| < 2b (ii) for every x ∈ R, |h(x)| ≤ Ce−2πb|x| Proof. In fact, b b b h(x) = λ(x) +b h(x)ψ(x)


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for all x ∈ R, from where it follows that the real singularities of H(z) :=

b λ(z) b 1 − ψ(z)

are removable. Our next aim is to show that H is a holomorphic extension of b h. Now if λ and ψ are supported by [−A, A], by Paleyb and ψb are holomorphic and for Wiener-Schwartz Theorem [22, 7.3.1] λ each N ∈ N there is CN > 0 such that b b |λ(z)| ≤ CN eA|Imz| (1 + |z|)−N and |ψ(z)| ≤ CN eA|Imz| (1 + |z|)−N . b From these estimates it is clear that |ψ(z)| < 1 for |Imz| < B and |z| > AB 1/N (CN e ) in particular if |Rez| > (CN eAB )1/N . From the analicity b ψb − 1 has only finitely many zeroes in the region |Imz| < B, of ψ, |Rez| < (CN eAB )1/N . Thus, by further reducing B, we may assume that ψb − 1 has only real zeroes in the strip |Imz| < B. Thus H has only real singularities in this strip. We take b = B/2 and by abuse of the notation we still denote by H the holomorphic function on |Imz| < 2b obtained when we remove all the real singularities. Then H is a holomorphic extension of b h, hence for any fixed x > 0 we have ∫ ∞ h(x) = H(t)e2πitx dt. −∞

Denote by γR the rectangle with vertices −R, R, R + ib, −R + ib. Then ∫ H(z)e2πizx dz = 0. γR

Moreover from the previous estimates |H(z)| < 2CN e2Ab (1 + |z|)−N for |z| > (CN e2Ab )1/N and |Imz| < 2b. Hence after estimating the integrals on the vertical segments and taking limits as R goes to infinity we deduce that ∫ ∞ b λ(t + ib) 2πi(t+ib)x e dt h(x) = b + ib) −∞ 1 − ψ(t and |h(x)| ≤ Ce−2πbx for some constant C > 0 independent of x. In the case x < 0 we can argue in the same way but with a rectangle in the lower half-plane and we finally get |h(x)| ≤ Ce−2πb|x| for every x ∈ R.


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Theorem 15. Let σ ∈ E 0 (R2 ) be a compactly supported distribution such that the convolution operator Tσ is hypoelliptic. Given f ∈ S(R) \ {0}, Qσ (f ) is not compactly supported. Proof. We suppose that Qσ (f ) is compactly supported. According to 2 b Lemma 13, f satisfies the hypothesis of Lemma 14, hence there is b b > 0 such that f (ω) ≤ e−2πb|ω| for all ω ∈ R. Then ∫ F (z) := fb(ω)e2πiωz dω R

is a holomorphic function in the strip |Imz| < b, hence f is a realanalytic function. On the other hand, by Lemma 13, there are test functions ψ, λ ∈ D(R) so that h = λ + h ∗ ψ, where h = |f |2 . Since h and also h ∗ ψ are real analytic functions we conclude that λ is identically zero. Taking Fourier transforms we b Consequently ψ(ω) b get b h = b h · ψ. = 1 whenever b h 6= 0. Then the b analytic continuation principle implies that ψ(ω) = 1 for all ω, which is a contradiction. As the Laplace operator ∆ is elliptic [22, 7.1.19], thus hypoelliptic [22, 11.1.10], the previous theorem permits to answer the question we raised in the introduction about the non-compacteness of the support of ∆W (f ). A more general result is given in Theorem 22. Corollary 16. Let f ∈ S(R) be a non-zero function. Then W (f ) is not an harmonic function outside a compact set. Our next aim is to show that the hypothesis in the previous result can be weakened. In particular we want to show that a linear partial differential operator applied to the Wigner distribution of a function f 6= 0 in the Schwartz class cannot produce a compactly supported function. The key is to give an appropriate extension of Lemma 13. We recall that any continuous function g on Rd dominated by a polynomial is a distribution g ∈ S 0 (Rd ). Lemma 17. Let σ ∈ S 0 (R2 ) such that σ b is a continuous function dominated by a polynomial and let µ1 , µ2 ∈ S 0 (R) be defined by µb1 (ξ) = σ b(ξ, 0), µb2 (ξ) = σ b(0, ξ). Assume that µ1 , µ2 are compactly supported and the associated convolution operators Tµj (j = 1, 2) are hypoelliptic. Then, there are test functions ψ1 , ψ2 , λ1 , λ2 ∈ D(R) such that |f |2 = λ1 + |f |2 ∗ ψ1 and |fb |2 = λ2 + |fb |2 ∗ ψ2 .

14


Proof. By hypothesis, χ := σ ∗ W (f ) ∈ D(R2 ). Now put

∫ ϕ(x) := R

∫

As

R

χ(x, y) dy, ϕ ∈ D(R).

W (f )(x, y) dy = |f (x)|2 , x ∈ R

we have d \ ϕ(ξ) b =χ b(ξ, 0) = σ b(ξ, 0)W (f )(ξ, 0) = µb1 (ξ)|f |2 (ξ), ξ ∈ R hence ϕ = µ1 ∗ |f |2 . Now, since Tµ1 is a hypoelliptic operator there are a > 0 and E ∈ D0 (R), which is a C ∞ function outside [−a, a] such that E ∗ µ1 = δ ([22, 16.6.5]). Fix Ψ ∈ D(R) equals one on a neighborhood of [−a, a]. Then (ΨE) ∗ µ1 = δ − (1 − Ψ)E ∗ µ1 . Since (1 − Ψ)E is a C ∞ function then we have ψ1 := (1 − Ψ)E ∗ µ1 ∈ D(R)

(6)

and also λ1 := (ΨE) ∗ µ1 ∗ |f |2 = (ΨE) ∗ ϕ ∈ D(R). Finally |f |2 = λ1 + ψ1 ∗ |f |2 .

The argument for |fb |2 is similar.

Now, we can proceed as in the proof of Theorem 15 to get the following. Theorem 18. Let σ ∈ S 0 (R2 ) be such that σ b is a continuous function dominated by a polynomial and let µ1 , µ2 ∈ S 0 (R) be defined by µb1 (ξ) = σ b(ξ, 0), µb2 (ξ) = σ b(0, ξ). Assume that µ1 , µ2 be compactly supported and that the associated convolution operators Tµj (j = 1, 2) are hypoelliptic. Given f ∈ S(R) \ {0}, Qσ (f ) is not compactly supported. Example 19. Let us recall that a quadratic time-frequency representation Qσ satisfies the marginal property if ∫ Qσ (f )(x, ω)dω = |f (x)|2 , x ∈ R R

and

∫ R

Qσ (f )(x, ω)dx = |fb(ω)|2 , ω ∈ R

for every f ∈ S(R). Of course, proper assumptions on σ should be made so that the integrals on the left hand sides make sense. As was shown in [8] for reasonable behaved σ, the marginal property is satisfied if and only if σ b(x, 0) = σ b(0, ω) = 1 for all x, ω ∈ R. If σ ∈ S 0 (R2 ) be a tempered distribution such that σ b is a continuous function and


15

σ b(x, 0) = σ b(0, ω) = 1 for all x, ω ∈ R.. Then σ satisfies the hypothesis in Theorem 18 with µ1 = µ2 = δ. In this case the conclusion of the Theorem was already known. Example 20. Let µ ∈ E 0 (R) be a compactly supported distribution and define ( ) σ := F −1 µ b(ax + by + h(xy)) ∈ S 0 (R2 ), where h is any continuous function of at most polynomial growth on R and vanishing at the origin. We suppose (1) Tµ is hypoelliptic and ab 6= 0 or (2) a = b = 0 and µ b(0) 6= 0. Then σ need not be compactly supported but satisfies the hypothesis of Theorem 18. Example 21. Let µ ∈ E 0 (R) be such that Tµ is hypoelliptic and take λ ∈ R with µ b(−λ) 6= 0. Take σ = W (µ, e−2πiλt ) hence σ ∈ / E 0 (R2 ) as follows from [14, 5.9]. From Theorem 18 we have that Qσ (f ) is not compactly supported unless f = 0. In fact, x

σ b(x, ω) = e2πiω(λ− 2 ) µ b(x − λ). Hence σ b(0, ω) = µ b(−λ)e2πiωλ , σ b(x, 0) = µ b(x − λ). We observe that Q(f ) = Specµ˜,e2πiλt (f, f ) (see [2] for the definition of the generalized spectrogram). We conclude with an application of our results to linear partial differential equations in time-frequency space. It is a consequence of Theorem 18 and, in our opinion, is the main result of this section. We prove that if P (Dx , Dy ) is a partial differential operator, then for f ∈ S(R), the Wigner function W (f ) cannot be solution of the equation P (Dx , Dy )u = v if the right hand side v has compact support. Theorem 22. Let P (Dx , Dy ) 6= 0 be a partial differential operator. Then, for f ∈ S(R) \ {0}, P (Dx , Dy )W (f ) is not compactly supported. Proof. In fact, let us assume that P (Dx , Dy )W (f ) is compactly supported. Now we factorize P (Dx , Dy ) = Dxn Dym Q(Dx , Dy ), where m, n ∈ N0 and Q(0, Dy ) and Q(Dx , 0) are non-zero differential operators. Since H := Q(Dx , Dy )W (f ) is in the Schwartz class and Dxn Dym H is compactly supported we get that also Q(Dx , Dy )W (f ) is compactly supported. Take σ ∈ E 0 (R2 ) \ {0} with σ b = Q and define µj according to Theorem 18. The convolution operators Tµj (j = 1, 2) are non-zero ordinary linear differential operators with constant coefficients, therefore they are hypoelliptic. Consequently f is identically zero.

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4. Conclusion Due to the uncertainty principle, some sort of restriction on the support of any time-frequency representation is to be expected and the type of restriction we are dealing with in this paper, e.g. compact support, is among the most natural and simple conditions. We have obtained several classes of kernels σ ∈ S 0 (R2d ) such that Qσ f cannot be compactly supported for f ∈ S(Rd ) unless f = 0. In contrast we observe that, according to Example 5, there are σ ∈ S 0 (R2d ) \ {0} and f ∈ S(Rd ) \ {0} such that Qσ (f ) = 0. In order to get a more clear idea of the extent of the obtained results it would be natural to look for examples of kernels and functions for which the corresponding Cohen representation has non trivial compact support. At the moment we do not have an answer whether such examples exist. Special emphasis is put in the case that d = 1 and σ is a finite linear combination of derivatives of the Dirac delta function. In this case Qσ f is given by the action of a partial differential operator P (Dx , Dy ) on the Wigner transform W (f ) of f and our main result, Theorem 22, proves that a linear partial differential operator applied to the Wigner distribution of a function f 6= 0 in the Schwartz class cannot produce a compactly supported function. In particular, W (f ) is not a harmonic function outside a compact set (Corollary 16). Theorem 22 is a particular case of the more general Theorem 18 where hypoellipticity of convolution operators plays an important role. We should mention that the idea in the proof of Theorem 18 was already contained in Theorem 15. We end the paper by some considerations and remarks about possible developments. We point out that the results of this paper can be viewed as a generalization to suitable subclasses of the Cohen class of more quantitative estimates obtained for some classical time-frequency representations. Namely we recall that for the ambiguity function ∫ A(f, g)(x, ω) = e−2πitω f (t + x/2)g(t − x/2)dt, f, g ∈ L2 (Rd ), Rd

estimates of the type ∫ kf |22 kgk22

≤ C(Σ)

R2d \Σ

|A(f, g)(x, ω)|2 dx dω

has been obtained in [13], where Σ ⊂ R2d forms a strong annihilating e with Σ e = {(ξ, η) : (−η, ξ) ∈ Σ)}. This leads to consider pair (Σ, Σ) inequalities for the Wigner representation of the type kW (f )kL2 (R2d ) ≤ C(Σ)kW (f )kL2 (R2d \Σ)

(7)


17

and for sets Σ with finite Lebesgue measure the constant C(Σ) can be estimated in terms of the measure of Σ, using the results in [24]. In view of the estimate (7) it seems a reasonable matter of research to investigate whether the results of this paper could be given in a more quantitative form of the type kQσ (f )kL2 (R2d ) ≤ C(K)kQσ (f )kL2 (R2d \K) for suitable subsets K ⊂ R2d and with possible estimation of the constant C(K). The authors want to thank the managing editor and the referees who helped in clarifying the presentation of the manuscript. Acknowledgement: This paper was finished during a visit of the last two authors to the University of Torino. They want to thank the Department of Mathematics for their hospitality. The research of C. Fernández and A. Galbis was supported by MEC and FEDER, Projects MTM2007-62643, MTM2010-15200 and net MTM2006-26627-E. References [1] Benedicks, M., On Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106 (1), 180–183 (1985) [2] Boggiatto, P., De Donno, G., Oliaro, A., Uncertainty principle, positivity and Lp -boundedness for generalized spectrograms. J. Math. Anal. Appl. 335 (1), 93–112 (2007) [3] Bonami, A., Demange, B., A survey on uncertainty principles related to quadratic forms. Collect. Math. Vol. Extra, 1–36 (2006) [4] Bonami, A., Demange, B., Jaming, Ph., Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19, 22–35 (2003) [5] de Bruijn, N.G., A theory of generalized functions, with applications to Wigner distributions and Weyl correspondence. Nieuw Archief voor Wiskunde 21, 205– 281 (1973) [6] Claasen, T.A.C.M., Mecklenbräuker, W.F.G., The Wigner distribution-a tool for time-frequency signal analysis. I. Continuous time signals. Philips J. Res. 35(3), 217–250 (1980) [7] Claasen, T.A.C.M., Mecklenbräuker,W.F.G. , The Wigner distribution-a tool for time-frequency signal analysis. II. Discrete time signals. Philips J. Res. 35(45), 276–300 (1980) [8] Claasen, T.A.C.M., Mecklenbräuker,W.F.G. , The Wigner distribution-a tool for time-frequency signal analysis. III. Relations with other time-frequency signal transformations. Philips J. Res. 35(6), 370–389 (1980) [9] Cohen, L., Time-Frequency Distributions - A Review. Proc. of IEEE, 77(7), 941-981 (1989) [10] Cohen, L., Time-Frequency Analysis. Prentice Hall Signal Proc. series, New Jersey (1995) [11] Cohen, L., The uncertainty principle for the short-time Fourier transform. Proc. Int. Soc. Opt. Eng. 22563, 80–90 (1995) [12] Dacorogma, B., Introduction to the Calculus of Variations. London: Imperial College Press (2004)

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[38] Wong, M.W., Wavelet Transform and Localization Operators. BirkhäuserVerlag, Basel (2002) P. Boggiatto: Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail address: [email protected] ńdez: Departamento de Ana ´lisis Matema ´tico, Universidad de C. Ferna Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain E-mail address: [email protected] ´lisis Matema ´tico, Universidad de VaA. Galbis: Departamento de Ana lencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain E-mail address: [email protected]