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.


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
Full Text: DOI arXiv EuDML


[1] Auslander, L. and Tolimieri, R.: Radar ambiguity functions and group theory. SIAM J. Math Anal. 16 (1985), 577-601. · Zbl 0581.43002 · doi:10.1137/0516043
[2] Bagchi, S.C. and Ray, S.K.: Uncertainty principles like Hardy’s theorem on some Lie groups. J. Austral. Math. Soc. Ser. A 65 (1999), 289-302. · Zbl 0930.22009
[3] de Bruijn, N.G.: A theory of generalized functions with applications to Wigner distribution and Weyl correspondence. Nieuw Arch. Wisk. (3) 21 (1973), 205-280. · Zbl 0269.46033
[4] de Buda, R.: Signals that can be calculated from their ambiguity function. IEEE Trans. Inform. Theory IT-16 (1970), 195-202. · Zbl 0192.56604 · doi:10.1109/TIT.1970.1054428
[5] Bueckner, H.F.: Signals having the same ambiguity functions. Techni- cal Report 67-C-456, General Electric, Research and Development Center, Schnectady, N.Y., 1967.
[6] Cohen, L.: Time-frequency distributions -a review. Proc. IEEE 77:7 (1989), 941-981.
[7] Cook, C.E. and Bernfeld, M.: Radar signals -an introduction to theory and application. Academic Press (1967).
[8] Cowling, M.G. and Price, J.F.: Generalizations of Heisenberg’s in- equality. In: Harmonic Analysis (eds. G. Mauceri, F. Ricci and G. Weiss), LNM, no. 992. pp. 443-449, Springer, Berlin, 1983. · Zbl 0516.43002
[9] Dembo, A., Cover, T.M. and Thomas, J.A.: Information-theoretic inequalities. IEEE Trans. Inform. Theory 37:6 (1991), 1501-1518. · Zbl 0741.94001 · doi:10.1109/18.104312
[10] Flandrin, P.: Separability, positivity, and minimum uncertainty in time- frequency energy distributions. J. Math. Phys. 39:8 (1998), 4016-4040. · Zbl 0928.94004 · doi:10.1063/1.532483
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.