# Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms.

Aline Bonami; Demange, Bruno, Jaming, Philippe

Revista Matemática Iberoamericana (2003)

- Volume: 19, Issue: 1, page 23-55
- ISSN: 0213-2230

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topBonami, Aline, and Demange, Bruno, Jaming, Philippe. "Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms.." Revista Matemática Iberoamericana 19.1 (2003): 23-55. <http://eudml.org/doc/39685>.

@article{Bonami2003,

abstract = {We extend an uncertainty principle due to Beurling into a characterization 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 positive 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 ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.},

author = {Bonami, Aline, Demange, Bruno, Jaming, Philippe},

journal = {Revista Matemática Iberoamericana},

keywords = {Incertidumbre; Transformada de Fourier; Transformada de Wigner; Funciones de variable compleja; Funciones holomorfas de varias variables; Función entera; Polinomios de Hermite; uncertainty principle; short-time Fourier transform; windowed Fourier transform; Gabor transform; ambiguity function; Wigner transform; spectrogram},

language = {eng},

number = {1},

pages = {23-55},

title = {Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms.},

url = {http://eudml.org/doc/39685},

volume = {19},

year = {2003},

}

TY - JOUR

AU - Bonami, Aline

AU - Demange, Bruno, Jaming, Philippe

TI - Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms.

JO - Revista Matemática Iberoamericana

PY - 2003

VL - 19

IS - 1

SP - 23

EP - 55

AB - We extend an uncertainty principle due to Beurling into a characterization 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 positive 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 ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

LA - eng

KW - Incertidumbre; Transformada de Fourier; Transformada de Wigner; Funciones de variable compleja; Funciones holomorfas de varias variables; Función entera; Polinomios de Hermite; uncertainty principle; short-time Fourier transform; windowed Fourier transform; Gabor transform; ambiguity function; Wigner transform; spectrogram

UR - http://eudml.org/doc/39685

ER -

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- Saifallah Ghobber, Dunkl-Gabor transform and time-frequency concentration

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