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

By Aline Bonami, B. Demange, Philippe Jaming. 2001

233 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/fbb6bc0e3750e227f1eaebb0ca3ab17c9f334b69

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In Applied Mathematics

Citations in the corpus

Uncertainty principles for the weinstein transform (2011, unknown): “A far reaching generalization of this result has been recently proved by Bonami, Demange and Jaming [6].”
Uncertainty principles for the Dunkl transform (2010, result): “Beurling’s theorem for the Dunkl transform Beurling’s theorem and Bonami, Demange, and Jaming’s extension are generalized for the Dunkl transform as follows.”
Optimal OFDM design for time-frequency dispersive channels (2003, methodology): “An alternative formulation is the following Heisenberg-type inequality [17], [ 18 ].”
The Hardy Uncertainty Principle Revisited (2010, background): “There has also been considerable interest in a better understanding of these results and on its extensions to higher dimensions, and to other settings ([2], [3], [14], [16] and [15]).”
Orthonormal Sequences in L2(Rd) and Time Frequency Localization (2009, background): “J Fourier Anal Appl (2010) 16: 983–1006 987 It is well known that the Hermite functions are extremal in many problems concerning the uncertainty principle, see for example [3, 10], and [14].”
Spaces of Test Functions via the STFT (2004, unknown): “In particular, if αi + βi < 1 for some i, then M p mλ = {0} for all λ > 0, and thus S β α = {0} [3] (compare Lemma 3.”