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Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. (English) Zbl 1037.42010
The authors extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ in $\bbfR^d$ which may be written as $P(x) \text{ exp} (-\langle Ax,x \rangle)$, with $A$ a real symmetric positive definite matrix, are characterized by integrability conditions on the product $f(x) \widehat{f}(y)$. They also 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). The paper ends with a sharp version of Heisenberg’s inequality for this transform.

MSC:
42B10Fourier type transforms, several variables
33C45Orthogonal polynomials and functions of hypergeometric type
94A12Signal theory (characterization, reconstruction, filtering, etc.)
42C15General harmonic expansions, frames
42C40Wavelets and other special systems
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References:
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