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Spectral theorems associated with the directional short-time Fourier transform

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Abstract

In this paper, we are interested in the directional short-time Fourier transform by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on \(L^{p}({\mathbb {R}}^{d})\), \(1 \le p \le \infty \). After wards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving \(\varepsilon \)-concentration of the generalized two-wavelet multipliers. Moreover we study functions whose time–frequency content are concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.

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Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. First author thanks professors K. Trimèche and M. W. Wong for their help.

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Correspondence to Hatem Mejjaoli.

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Mejjaoli, H., Omri, S. Spectral theorems associated with the directional short-time Fourier transform. J. Pseudo-Differ. Oper. Appl. 11, 15–54 (2020). https://doi.org/10.1007/s11868-019-00308-z

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Keywords

  • Directional short-time Fourier transform
  • Generalized Landau–Pollak–Slepian operator
  • Donoho–Stark type uncertainty principle
  • Approximation inequalities
  • Generalized multipliers
  • Generalized two-wavelet multipliers
  • Schatten–von Neumann class
  • \(L^{p}\)-boundedness
  • \(L^{p}\)-compactness

Mathematics Subject Classification

  • Primary 47G10
  • Secondary 42B10
  • 47G30