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Uncertainty principles for time-frequency representations. (English) Zbl 1039.42004

Feichtinger, Hans G. (ed.) et al., Advances in Gabor analysis. Basel: Birkhäuser (ISBN 0-8176-4239-0/hbk). Applied and Numerical Harmonic Analysis, 11-30 (2003).
A Fourier uncertainty principle is a statement regarding the limitation of joint time-frequency localization of a nontrivial function \(f\) and its Fourier transform \(\hat f\). Typical forms include Heisenberg’s inequality: \(\| x f(x)\| _{L^2} \| \xi\hat f(\xi)\| _{L^2}\geq \| f\| _{L^2}^2/(4\pi)\); Benedick’s theorem: a function and its Fourier transform cannot both live on sets of finite Lebesgue measure; and Hardy’s theorem: \(\| f(x) e^{\alpha\pi x^2}\| _{L^\infty} \| \hat f(\xi) e^{\pi \xi^2/\alpha}\| _{L^\infty} < \infty\) with \(\alpha > 0\) implies \(f\) is a multiple of \(e^{-\alpha\pi x^2}\). There are many variations on these basic themes – each of which comes with their own sets of techniques – in the literature. Here, the author systematically reviews the discoveries that each such Fourier uncertainty principle has a corresponding formulation in terms of time-frequency distributions such as the short-time Fourier transform (STFT) or Wigner distribution. Proofs of analogues of Benedicks’ and Hardy’s theorems and formulations of corresponding statements about essential time-frequency localization are provided in the case of the STFT. Importantly, the author elucidates the theme underlying the methods of these proofs, namely, that statements for the STFT are always reduced to their counterpart statements for \(f\) and \(\hat f\) by a suitable use of symmetry properties of the time-frequency plane. In the case of the STFT these symmetries are captured in terms of certain identities that the STFT satisfies. As such, the extent to which uncertainty principles for the pair \((f,\hat f)\) can be extended to more general Cohen class time-frequency distributions remains unclear.
For the entire collection see [Zbl 1005.00015].

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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