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[BDJam] Revista Matematica Iberoamericana 19 (2003) 23–55. Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms Aline BONAMI, Bruno DEMANGE & Philippe JAMING Abstract : We extend an uncertainty principle due to Beurling into a char- acterization of Hermite functions. More precisely, all functions f on Rd which may be written as P (x) exp(Ax, x), with A a real symmetric definite pos- itive matrix, are characterized by integrability conditions on the product f(x)f?(y). We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambigu- ity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform. Keywords : Uncertainty principles; short-time Fourier transform; windowed Fourier transform; Gabor transform; ambiguity function; Wigner transform; spectrogramm. AMS subject class : 42B10;32A15;94A12. 1. Introduction and Notations. Uncertainty principles state that a function and its Fourier transform cannot be simulta- neously sharply localized. To be more precise, let d ≥ 1 be the dimension, and let us denote by ?., .? the scalar product and by ?.? the Euclidean norm on Rd. Then, for f ? L2(Rd), define the Fourier transform of f by f?(y) = ∫ Rd f(t)e?2ipi?t,y?dt.

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  • beurling-hormander

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[BDJam]
RevistaMatema´ticaIberoamericana19(2003)2355.
Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms
Aline BONAMI, Bruno DEMANGE & Philippe JAMING
Abstract :We extend an uncertainty principle due to Beurling into a char-acterization of Hermite functions. More precisely, all functionsfonRdwhich may be written asP(x) exp(Ax, x), withAa real symmetric definite pos-itive matrix, are characterized by integrability conditions on the product b f(x)f(y). We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambigu-ity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg’s inequality for this transform. Keywords :Uncertainty principles; short-time Fourier transform; windowed Fourier transform; Gabor transform; ambiguity function; Wigner transform; spectrogramm. AMS subject class :42B10;32A15;94A12.
1.Introduction and Notations.
Uncertainty principles state that a function and its Fourier transform cannot be simulta-neously sharply localized. To be more precise, letd1 be the dimension, and let us denote byh., .ithe scalar product and byk.kthe Euclidean norm onRd for. Then,fL2(Rd), define the Fourier transform offby fb(y)ZRd2ht,yidt. =f(t)e The most famous uncertainty principle, due to Heisenberg and Weil, can be stated in the following directional version : Heisenberg’s inequality. Leti= 1, . . . , dandfL2(Rd). Then 2 (1)ainfRZRd(xia)2|f(x)|2dxbinfRZRd(ξib)2fb(ξ)2dξ≥ kfk4 16π2. Moreover (1) is an equality if and only iffis of the form f(x) =C(x1, . . . , xi1, xi+1, . . . , xn)e2iπbxieα(xia)2 whereCis a function inL2(Rd1),α >0, andaandbare real constants for which the two infimums in (1) are realized. The usual non-directional uncertainty principle follows easily from this one. We refer to the recent survey articles by Folland and Sitaram [FS] and Dembo, Cover and Thomas [DCT] aswellasthebookofHavinandJ¨oricke[HJ]forvariousuncertaintyprinciplesofdierent nature which may be found in the literature. One theorem stated in [FS] is due to Beurling. ItsproofhasbeenwrittenmuchlaterbyH¨ormanderin[Ho].Ourrstaimistoweakenthe 151
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assumptions so that non zero solutions given by Hermite functions are also possible. More precisely, we will prove the following theorem : Theorem 1.1ng-Hurliande¨orm)etrpyeB(. LetfL2(Rd)andN0. Then b (2)Z ZRd×Rd(1|+fk(xx)k||+f(kyy)|k)Ne2π|hx,yi|dxdy <+if and only iffmay be written as f(x) =P(x)eπhAx,xi , whereAis a real positive definite symmetric matrix andPis a polynomial of degree<2.
In particular, forNd, the functionfsidineitinalsregiroro¨Hdnamrleug-inllca.By0 theorem is the above theorem ford= 1 andN= 0. An extension tod1 but stillN= 0 has been given, first by S.C. Bagchi and S. K. Ray in [BR] in a weaker form, then very recently by S. K. Ray and E. Naranayan in the present form. Their proof, which relies on the one dimensional case, uses Radon transform [RN]. Let us remark that the idea of characterizing Hermite functions by pointwise vanishing at infinity, for both the function and its Fourier transform, goes back to Hardy. Indeed, such a characterization is contained in Hardy’s original theorem [Ha], (though textbooks usually restrict attention to the characterization of gaussians in Hardy’s Theorem). One may also consult [Pf] for extensions. The proof given here with integrability conditions uses new ingredientscomparedtotheoriginalproofofHo¨rmander[Ho].Atthesametime,itsimplies Ho¨rmandersargumentforthecaseN= 0,d= 1, in such a way that the proof can now be given in any textbook on Fourier Analysis. We give this last one in the Appendix, since it may be useful in this context. The previous theorem has as an immediate corollary the following characterization. Corollary 1.2. A functionfL2(Rd)may be written as f(x) =P(x)eπhAx,xi, withAa real positive definite symmetric matrix andPa polynomial, if and only if the function b f(x)f(y) exp(2π|hx, yi|) is slowly increasing onRd×Rd. As an easy consequence of the previous theorem, we also deduce the following corollary, which generalizes the Cowling-Price uncertainty principle (see [CP]). Theorem 1.3(Cowling-Price type). LetN0. Assume thatfL2(Rd)satisfies ZRd|f(x)|(1e+πa||xxjj||2)Ndx <+a dZRd|fbeπb|yj|2+n (y)|(1 +|yj|)Ndy < forj= 1,∙ ∙ ∙, dand for some positive constantsaandbwithab= 1. Thenf(x) = P(x)akxk2for some polynomialP. e