HARDY S THEOREM AND THE SHORT-TIME FOURIER TRANSFORM OF SCHWARTZ FUNCTIONS

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1 This preprint is in final form. It appeared in: J. London Math. Soc , pp c London Mathematical Society HARDY S THEOREM AND THE SHORT-TIME FOURIER TRANSFORM OF SCHWARTZ FUNCTIONS KARLHEINZ GRÖCHENIG, GEORG ZIMMERMANN Abstract. We characterize the Schwartz space of rapidly decaying test functions by the decay of the short-time Fourier transform or cross-wigner distribution. Then we prove a version of Hardy s theorem for the short-time Fourier transform and for the Wigner distribution. 1. Introduction In classical harmonic analysis, uncertainty principles are inequalities or vanishing theorems which involve a function f and its Fourier transform f simultaneously. Uncertainty principles give a precise meaning to the generic statement that a function and its Fourier transform cannot both be small. See, e.g., [4, 5, 6, 12] for uncertainty principles of this type. In quantum mechanics and in signal analysis, uncertainty principles are often discussed for simultaneous time-frequency representations, such as the short-time Fourier transform STFT or the Wigner distribution. Such time-frequency representations are transforms which map a function f on to a function V f on, the so-called phase space or time-frequency plane. In this context, uncertainty principles say that V f cannot have arbitrarily large peaks or that its essential support has a volume of minimum size 1. Notable examples of this line of thought are Janssen s work on the positivity of the Wigner distribution [16], the radar uncertainty principle [8], Lieb s inequalities [18], or the recent improvement of the Cowling-Price inequalities in terms of the STFT [10]. For a generous survey of uncertainty principles see [9]. The underlying philosophy of our contribution to uncertainty principles is the conviction that every uncertainty principle about f and f can be translated into an inequality about the STFT or the Wigner distribution and vice versa. As a non-trivial manifestation of this metatheorem we prove a version of Hardy s Theorem for the STFT and the Wigner distribution. Hardy s Theorem is a precise statement of how fast a function and its Fourier transform can decay at best. As so often, Gaussian functions play a central role. Theorem 1.1 Hardy s uncertainty principle. [6, 9, 11] Let f L 2, and assume that 1.1 f x = Oe aπx2 and f ξ = Oe bπξ Mathematics Subject Classification. 42B10,46F05,94A12. Key words and phrases. Gauss function, uncertainty principle, Wigner distribution, short-time Fourier transform, Hardy s theorem. Both authors would like to thank the members of the NuHAG Numerical Harmonic Analysis Group at the Department of Mathematics, University of Vienna, for their hospitality. 1

2 2 KARLHEINZ GRÖCHENIG, GEORG ZIMMERMANN for some constants a, b > 0 with a b = 1. Then for some c C. f x = c e aπx2 We show an analogous statement for the STFT. Let g S be a fixed window function. Then the STFT of f S with respect to g is defined to be the function on phase space given by 1.2 V g fx, ξ = f t gt x e 2πiξt dt. For the Gaussian f t = e πt2, we will see that V f fx, ξ = 2 d/2 e π x2 +ξ 2 /2 e πiξx. It turns out that this is the optimal decay rate for the STFT. In fact, we will prove the following. Theorem 1.2. Let g, f S S, and assume that 1.3 V g fx, ξ = Oe πx2 +ξ 2 /2, and that V g f does not vanish identically. Then V g fx, ξ = C e 2πiζ 0x ξz 0 e π x2 +ξ 2 /2 e πiξx for some z 0, ζ 0, and f and g are multiples of e 2πiζ 0t e πt z 0 2. A similar statement holds for the cross- Wigner distribution of f and g. As a further argument for our metatheorem one might quote Benedicks Theorem. Benedicks [1] showed that if f L 1 satisfies λsupp f λsupp f <, where λ denotes Lebesgue measure on, then f 0. As a by-product of our approach we will give a very short proof for the corresponding version for the STFT due to Jaming, Janssen, and Wilczock [14, 17, 21]. Theorem 1.3. If λsupp V g f <, then V g f 0, so f 0 or g 0. Hardy s Theorem for the STFT is intrinsically related to the definition of Fourier invariant spaces of test functions. Test functions are defined by smoothness and decay conditions so that the resulting Fréchet space is invariant under the Fourier transform. This invariance then allows to define and study the Fourier transform on the dual space, a Fréchet space of distributions. To put Hardy s Theorem for the STFT in perspective we study the STFT of Schwartz functions and derive a characterization of the Schwartz class in terms of the decay of the STFT. As a result, it seems that S may be conveniently introduced and studied by its time-frequency properties. In this context, Hardy s Theorem represents an essential limitation for the decay of test functions in the time-frequency plane. The paper is organized as follows. First we list the notation used and remind the reader of the main properties of the STFT. In Section 2, we characterize functions in S by the decay of their STFT. Our main result Theorem 1.2 and some consequences are proved in Section 3.

3 HARDY S THEOREM AND THE STFT 3 Notation. For the Fourier transformation we use the normalization f τ = f t e 2πiτt dt. Then Parseval s identity f, g = f, ĝ holds for all f, g L 2, and the Gaussian function g 0 t = e πt2 is invariant under the Fourier transform, i.e., ĝ 0 = g 0. The involution gt = g t has the property g = ĝ. Translation and modulation are defined by T x gt = gt x and M ξ gt = e 2πiξt gt, their composition is the time-frequency shift Its Fourier transform is given by M ξ T x gt = e 2πiξt gt x. M ξ T x g = e 2πiξx M x T ξ ĝ. With this notation the short-time Fourier transform STFT can be written as V g fx, ξ = f t gt x e 2πiξt dt = f T x gξ = f, M ξ T x g. The main properties of the STFT are collected in the following lemma. Lemma 1.4. For functions f, f i, g, g i, and h S, the STFT has the following properties. i Switching f and g ii Inversion formula V f gx, ξ = e 2πiξx V g f x, ξ. b V g fx, ξ M ξ T x h dxdξ = h, g f. iii Orthogonality relations V g1 f 1, V g2 f 2 L 2 b = f 1, f 2 L 2 g 1, g 2 L 2. iv STFT of time-frequency shifts Vg M ζ T z f x, ξ = e 2πiξ ζz V g fx z, ξ ζ. v VMζ T z gm ζ T z f x, ξ = e 2πiζx ξz V g fx, ξ. vi Fourier transform of a product of STFTs Vg1 f 1 V g2 f 2 y, η = Vf2 f 1 V g2 g 1 η, y. Proof. ii V g fx, ξ M ξ T x ht dxdξ = R bd f Tx gξ e 2πiξt dξ ht x dx R bd = f t gt x ht x dx = h, g f t.

4 4 KARLHEINZ GRÖCHENIG, GEORG ZIMMERMANN iii V g1 f 1, V g2 f 2 L 2 R bd R = V g1 f 1 x, ξ f 2 t g 2 t x e 2πiξt dt dxdξ d R bd R d = V g1 f 1 x, ξ g 2 t x e 2πiξt dxdξ f 2 t dt R bd = g 2, g 1 f 1 t f 2 t dt = f 1, f 2 L 2 R ii d g 1, g 2 L 2. R d vi Vg1 f 1 V g2 f 2 y, η = V g1 f 1 x, ξ V g2 f 2 x, ξ e 2πixy+ξη dxdξ R bd Vg1 f 1, V My T η g 2 M y T η f 2 = v = iii f1, M y T η f 2 g1, M y T η g 2 = Vf2 f 1 V g2 g 1 η, y. The remaining properties are verified by straighforward calculations and left to the reader. Lemma 1.5. If g 0 t = e πt2 and f = g = M ζ0 T z0 g 0, then V g fx, ξ = 2 d/2 e 2πiζ 0x ξz 0 e πiξx e πx2 +ξ 2 /2. Proof. For f = g = g 0, we obtain V g fx, ξ = e πt2 e πt x2 e 2πiξt dt R d = e πs+x/22 e πs x/22 e 2πiξs+x/2 ds = e πiξx e πx2 /2 e 2πiξs ds e 2πs2 = 2 d/2 e πiξx e πx2 +ξ 2 /2. The general case follows by Lemma 1.4.v. 2. The characterization of the Schwartz space The Schwartz space S of rapidly decaying test functions is the Fréchet space of functions on generated by the family of seminorms { A = X β D α f } L : α, β N d 0. Here we use the multi-index notation d D α α j f := t α f j j j=1 and X β f t := d j=1 t β j j f t, where α, β N d 0. Furthermore, α = d j=1 α j, and α β means that α j β j for j = 1,..., d. We start with an algebraic lemma for interchanging the operators X β D α and M ξ T x.

5 Lemma 2.1. For g S, we have X β D α M ξ T x g = δ β HARDY S THEOREM AND THE STFT 5 β δ α γ x δ 2πiξ γ M ξ T x X β δ D α γ g. Proof. Obviously, we have X β T x g t = t β gt x = T x t+x β gt = T x x + X β g t 2.1 = T x δ β β δ x δ X β δ g t = δ β β δ x δ T x X β δ g t. Furthermore, Leibniz s rule implies D α M ξ h t = D α e 2πiξt ht = α γ D γ e 2πiξt D α γ ht 2.2 = α γ 2πiξ γ M ξ D α γ h t. Combining 2.1 and 2.2 yields X β D α M ξ T x g = X β α γ 2πiξ γ M ξ D α γ T x g = α γ 2πiξ γ M ξ X β T x D α γ g = α γ 2πiξ γ M β ξ δ x δ T x X β δ D α γ g δ β α β γ δ 2πiξ γ x δ M ξ T x X β δ D α γ g. = δ β The following statement shows how to construct Schwartz functions as superpositions of time-frequency shifts. Proposition 2.2. Let g S be fixed. Assume that F : R 2d C has rapid decay, i.e., that for all n 0, there is a constant C n > 0 such that F x, ξ C n 1 + x + ξ n. Then the integral 2.3 f t := F x, ξ M ξ T x gt dxdξ R 2d defines a function f in S. Proof. The integral in 2.3 is absolutely convergent in t. Thus we may differentiate under the integral sign as long as the resulting integral is absolutely convergent, uniformly on

6 6 KARLHEINZ GRÖCHENIG, GEORG ZIMMERMANN compact sets. The latter is certainly true by virtue of the assumptions on g and F. Thus we obtain with Lemma 2.1 X β D α f t = F x, ξ X β D α M ξ T x g t dxdξ so = δ β X β D α f L δ β C β α F x, ξ x δ 2πiξ γ M δ γ ξ T x X β δ D α γ g t dxdξ, β α F x, ξ x δ 2πiξ γ X β δ D α γ g δ γ L dxdξ F x, ξ P x, ξ dxdξ, where C = max δ β, X β δ D α γ g L and P x, ξ = δ β β α δ γ x δ 2πiξ γ = d 1+ x j β j 1 + 2πξ j α j. The assumption on F implies X β D α f L <, and consequently f S. Theorem 2.3. Let g S \{0} be fixed. Then for f S, the following are equivalent: i f S. ii V g f SR 2d. iii For all n 0, there is C n > 0 such that V g fx, ξ C n 1 + x + ξ n x, ξ R 2d. Proof. i ii. Given f, g S, the function F x, t = f t gt x is in SR 2d. Since S is invariant under partial Fourier transformations, we obtain that V g f x, ξ = F x, t e 2πiξt dt = F t F is also in SR 2d. ii iii. Obvious. iii i. Set f # = g 2 2 V R 2d g fx, ωm ω T x g dx dω. Proposition 2.2 implies that f # S. At the same time the inversion formula for the STFT Lemma 1.4 ii shows that f # = f. Thus f S. Remark. i Theorem 2.3 can be considered folklore, however, we do not know any explicit references. [8] and [15] are closest in spirit, the result also follows from an abstract result about the smoothness of square integrable representations of nilpotent Lie groups [3]. ii With slightly more effort, we can also prove results similar to Theorem 2.3 for Björck s spaces of ultra-rapidly decaying test functions. For the theory of such test functions see [2, 20]. iii The equivalence of properties ii and iii in Theorem 2.3 shows that the decay conditions on V g f alone imply its smoothness. Thus a local property of the STFT is determined by its global behavior. This phenomenon also occurs for other function spaces, for instance for Feichtinger s algebra S 0, see [7]. j=1

7 HARDY S THEOREM AND THE STFT 7 3. The uncertainty principle In this section we prove Theorem 1.2 and some consequences. Our goal is to show that the Gaussian decay 3.1 V g fx, ξ = Oe πx2 +ξ 2 /2 of a non-zero STFT implies that both f and g are multiples of the same time-frequency shift M ζ0 T z0 g 0. For the proof, we will make use of an auxiliary function with a remarkable invariance property. Lemma 3.1 [14]. For f, g S, the function F x, ξ = e 2πiξx V g f x, ξ V g f x, ξ satifies F y, η = F η, y. Proof. Note that by Lemma 1.4.i, so by Lemma 1.4.vi, F x, ξ = V g f x, ξ V g f x, ξ e 2πiξx = V g f V f g x, ξ, F y, η = V g f V f g y, η = V g f V f g η, y = F η, y. Proof of Theorem 1.2. Theorem 2.3 combined with the decay condition 3.1 implies that f S, so we may apply the rules from Lemma 1.4. For z, ζ, consider the function 3.2 F z,ζ x, ξ = e 2πiξx e 2πiξ ζz V g fx z, ξ ζ e 2πi ξ ζz V g f x z, ξ ζ. By Lemma 1.4.iv, F z,ζ x, ξ = e 2πiξx V g M ζ T z f x, ξ V g M ζ T z f x, ξ, so by Lemma 3.1, F z,ζ y, η = F z,ζ η, y. The assumption 3.1 yields F z,ζ x, ξ C 2 e π/2x z2 +ξ ζ 2 e π/2 x z 2 + ξ ζ 2 = C z,ζ e πx2 +ξ 2, and thus we also have F z,ζ y, η = F z,ζ η, y C z,ζ e πy2 +η 2. By Hardy s Theorem e.g., see [11], this implies 3.3 F z,ζ x, ξ = C z,ζ e πx2 +ξ 2, where 3.4 C z,ζ = F z,ζ 0, 0 = e 4πiζz V g f 2 z, ζ.

8 8 KARLHEINZ GRÖCHENIG, GEORG ZIMMERMANN By assumption, there exists z, ζ such that C z,ζ = V g f z, ζ 2 0. For this, we obtain F z,ζ x, ξ = V g fx z, ξ ζ V g f x z, ξ ζ = C z,ζ e πx2 +ξ 2 0 x, ξ, i.e., in particular, V g fx, ξ 0 for all x, ξ. Thus we may define Hx, ξ := log V g f x, ξ and, since is simply connected, we obtain a well-defined continuous function H : C after choosing some branch of the logarithm at H0, 0. Combining 3.2, 3.3, and 3.4 yields e 4πiζz V g f 2 z, ζ e πx2 +ξ 2 = = e 2πiξx e 2πiξ ζz V g fx z, ξ ζ e 2πi ξ ζz V g f x z, ξ ζ, from which we obtain by taking the logarithm 4πiζz + 2Hz, ζ πx 2 + ξ 2 = = 2πiξx 2πiξ ζz + Hz x, ζ ξ 2πi ξ ζz + Hz+x, ζ+ξ + 2πik for some k Z. Letting x, ξ = 0, 0 shows that k = 0, and we obtain Hz, ζ+x, ξ 2Hz, ζ + Hz, ζ x, ξ = πx 2 +ξ 2 2πiξx. This is an inhomogeneous linear difference equation, for which we can guess the solution H 0 x, ξ = π x 2 +ξ 2 /2 πiξx, and thus by Lemma 3.2 below, the general solution is given by Hx, ξ = π x 2 +ξ 2 /2 πiξx + ax + bξ + c for some a, b C d, c C. This is equivalent to and thus V g fx, ξ = C e π x2 +ξ 2 /2 πiξx ax bξ Vg fx, ξ = C e π x 2 +ξ 2 /2 Reax Rebξ. But this satisfies 3.1 if and only if Rea = Reb = 0. Therefore, we may define ζ 0 = a/2πi and z 0 = b/2πi and obtain for V g f the form claimed in the theorem. The inversion formula stated in Lemma 1.4.ii implies that V g f determines f up to a multiplicative constant, and because of Lemma 1.4.i, the same is true for g. Therefore, it suffices to show that if f = g = M ζ0 T z0 g 0, then V g f does indeed have the above form which was done in Lemma 1.5. Lemma 3.2. The continuous solutions of the homogeneous linear difference equation f x+y 2f x + f x y = 0 on are given by f x = a x + c for some a C d, c C.

9 HARDY S THEOREM AND THE STFT 9 Proof. By splitting f into its real and imaginary parts, we may assume that f is realvalued. The case d = 1 is well known and can be shown, e.g., using the theory of convex functions. For d > 1, it is immediate that f is affine along any straight line in by reduction to the case d = 1. Considering any three non-collinear points in, we obtain that f is affine on any triangle and thus on any plane. This extends inductively. Corollary 3.3. Let g, f S S, and assume that 3.5 V g fx, ξ = Oe πa x2 +b ξ 2 /2 for some constants a, b > 0. Then three cases can occur. i If a b = 1 and V g f 0, then both f and g are multiples of a time-frequency shift of the Gaussian e aπt2. ii If a b > 1, then V g f 0, so f 0 or g 0. iii If a b < 1, then 3.5 is satisfied whenever f and g are suitably scaled finite linear combinations of Hermite functions. Proof. i Set ρ = b/a 1/4, f ρ t = ρ d/2 f ρt, and g ρ t = ρ d/2 gρt. A short calculation shows that Vgρ f ρ x, ξ = V g fρx, ξ/ρ = Oe ab πx 2 +ξ 2 /2, so the statement is an immediate consequence of Theorem 1.2. ii If a b > 1 and V g f 0, then 3.5 is also satisfied for a b = 1. Using i and a dilation, we obtain that V g fx, ξ = e πax2 + 1 a ξ2 /2, contradicting the assumption ab > 1. Therefore V g f 0, and thus by Lemma 1.4.ii either g 0 or f 0. iii By an explicit computation [8, 13, 19], the STFT of the Hermite functions H k and H n is given by VHk H n x, ξ = Lk,n πx 2 +ξ 2 e πx2 +ξ 2 /2, where the L k,n are Laguerre polynomials. The statement follows after applying a suitable dilation. Remark. We have stated this uncertainty principle in terms of the STFT. In signal analysis, a number of other time-frequency distributions is commonly used. Examples are the cross-ambiguity function sometimes also called Fourier-Wigner distribution Af, gx, ξ = f t+x/2 gt x/2 e 2πiξt dt along with the ambiguity function Af = Af, f; and the cross-wigner distribution W f, gx, ξ = f x+t/2 gx t/2 e 2πiξt dt along with the Wigner distribution W f = W f, f. Simple substitutions in the integral show that Af, gx, ξ = e πiξx V g fx, ξ, and W f, gx, ξ = 2 d e 4πiξx V g f2x, 2ξ,

10 10 KARLHEINZ GRÖCHENIG, GEORG ZIMMERMANN where g t = g t. Thus we can restate Corollary 3.3 immediately for the cross- ambiguity function by simply replacing V g f in 3.5 by Af, g. For the cross- Wigner distribution, Condition 3.5 needs to be replaced by W f, gx, ξ = Oe 2πa x2 +b ξ 2, then the claim of the corollary holds as stated. Finally, we prove Theorem 1.3. V g f 0. We have to show that if λsupp V g f <, then Proof of Theorem 1.3. For z, ζ, consider F z,ζ as defined in 3.2. The assumption on V g f implies that the support of F z,ζ has finite measure, and by Lemma 3.1, the same is true for F z,ζ. Thus we may apply Benedicks result [1] to obtain that F z,ζ 0. Since this holds for all z, ζ, and since we obtain V g f 0. F z, ζ 0, 0 = e 4πiζz V g fz, ζ 2, References [1] M. Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl [2] G. Björck, Linear partial differential operators and generalized distributions, Ark. Mat [3] L. Corwin, Matrix coefficients of nilpotent Lie groups, Lie group representations III ed R. Herb, Lecture Notes in Math Springer, Berlin, 1984, pp [4] M. G. Cowling and J. F. Price, Bandwidth versus time concentration: the Heisenberg Pauli Weyl inequality, SIAM J. Math. Anal [5] D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math [6] H. Dym and H. P. McKean, Fourier series and integrals, Probab. Math. Statist. 14 Academic Press, Boston, [7] H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, Gabor Analysis and Algorithms: Theory and Applications eds H. G. Feichtinger and T. Strohmer, Appl. Numer. Harmon. Anal. Birkhäuser, Boston, 1998, pp [8] G. B. Folland, Harmonic analysis on phase space, Ann. of Math. Stud. 121 Princeton Univ. Press, Princeton, [9] G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl [10] K. Gröchenig, An uncertainty principle related to the Poisson summation formula, Studia Math [11] G. H. Hardy, A theorem concerning Fourier transforms, J. London Math. Soc [12] V. P. Havin and B. Jöricke, The uncertainty principle in harmonic analysis, Ergeb. Math. Grenzgeb Springer, Berlin, [13] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal [14] P. Jaming, Principe d incertitude qualitatif et reconstruction de phase pour la transformée de Wigner, C. R. Acad. Sci. Paris Sér. I Math [15] A. J. E. M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl [16] A. J. E. M. Janssen, Positivity properties of phase-plane distribution functions, J. Math. Phys

11 HARDY S THEOREM AND THE STFT 11 [17] A. J. E. M. Janssen, Proof of a conjecture on the supports of Wigner distributions, J. Fourier Anal. Appl [18] E. H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys [19] W. Schempp, Radar ambiguity functions, the Heisenberg group, and holomorphic theta series, Proc. Amer. Math. Soc [20] H. Triebel, Theory of function spaces, Monographs Math. 78 Birkhäuser, Basel, [21] E. Wilczock, Zur Funktionalanalysis der Wavelet- und Gabortransformation, Thesis, TU München, Karlheinz Gröchenig Department of Mathematics U 9 University of Connecticut Storrs, CT , USA address: groch@math.uconn.edu Georg Zimmermann Institut für Angewandte Mathematik und Statistik Universität Hohenheim Stuttgart, Germany address: gzim@uni-hohenheim.de