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

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.

[1]  K. Gröchenig,et al.  Hardy's Theorem and the Short‐Time Fourier Transform of Schwartz Functions , 2001 .

[2]  P. Jaming,et al.  Phase retrieval techniques for radar ambiguity problems , 1999 .

[3]  S. K. Ray,et al.  Uncertainty principles like Hardy's theorem on some Lie groups , 1998, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[4]  A. Janssen.  Proof of a conjecture on the supports of Wigner distributions , 1998 .

[5]  Philippe Jaming,et al.  Principe d'incertitude qualitatif et reconstruction de phase pour la transformée de Wigner , 1998 .

[6]  Patrick Flandrin,et al.  Separability, positivity, and minimum uncertainty in time-frequency energy distributions , 1998 .

[7]  G. Folland,et al.  The uncertainty principle: A mathematical survey , 1997 .

[8]  V. Havin.  The Uncertainty Principle in Harmonic Analysis , 1994 .

[9]  L. Hörmander.  A uniqueness theorem of Beurling for Fourier transform pairs , 1991 .

[10]  Amir Dembo,et al.  Information theoretic inequalities , 1991, IEEE Trans. Inf. Theory.

[11]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.

[12]  R. Tolimieri,et al.  Radar Ambiguity Functions and Group Theory , 1985 .

[13]  Rudolf de Buda,et al.  Signals that can be calculated from their ambiguity function , 1970, IEEE Trans. Inf. Theory.

[14]  B. Levin,et al.  Distribution of zeros of entire functions , 1964 .

[15]  G. W. Morgan.  A Note on Fourier Transforms , 1934 .

[16]  G. Hardy.  A Theorem Concerning Fourier Transforms , 1933 .

[17]  Elke Wilczok,et al.  New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform , 2000, Documenta Mathematica.

[18]  J. Poly,et al.  Z eros de fonctions holomorphes et contre-exemples en th eorie des radars , 2000 .

[19]  Christian Pfannschmidt.  A Generalization of the Theorem of Hardy: A Most General Version of the Uncertainty Principle for Fourier Integrals , 1996 .

[20]  Calvin H. Wilcox,et al.  The Synthesis Problem for Radar Ambiguity Functions , 1991 .

[21]  Augustus J. E. M. Janssen,et al.  Optimality property of the Gaussian window spectrogram , 1991, IEEE Trans. Signal Process..

[22]  M. Cowling,et al.  Generalisations of Heisenberg's inequality , 1983 .

[23]  de Ng Dick Bruijn.  A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence , 1973 .

[24]  Charles E. Cook,et al.  Radar Signals: An Introduction to Theory and Application , 1967 .

[25]  J. Poly,et al.  Zéros de fonctions holomorphes et contre-exemples en théorie des radars , 2022 .