Time-frequency localization operators: A geometric phase space approach. (English) Zbl 0672.42007

Daubechies, Ingrid

IEEE Trans. Inf. Theory 34, No. 4, 605-612 (1988).

The author defines a set of operators which localize both in time and frequency, similar but different from the prolate spheroidal functions, the time-frequency plane being treated as a whole (phase space). It is shown that for some simple domains in the time-frequency plane the associated localization operators are very simple having their eigenfunctions as Hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions. The operators are defined using the associated coherent state \(\phi_{p,q}\) of the point \((p,q)\in {\mathbb{R}}^ n\times {\mathbb{R}}^ n\) of the form: \(\phi_{p,q}(x)=e^{ipx}\phi (x-q),\) \(\phi \in L^ 2({\mathbb{R}}^ n)\).

Reviewer: L.Goras

MSC:

42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
33E10	Lamé, Mathieu, and spheroidal wave functions
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

spheroidal functions; phase space; localization operators; eigenfunctions; Hermite functions; incomplete gamma functions