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Niveau: Supérieur, Licence, Bac+2

[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.

[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.

- uncertainty principles
- ity function
- has been
- fourier transform
- beurling-hormander
- uncertainty principles state
- both results

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