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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 \(\mathbb{R}^d\) 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:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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References:

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