Banach Journal of Mathematical Analysis

A variant of the Hankel multiplier

Saifallah Ghobber

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Abstract

The first aim of this article is to survey and revisit some uncertainty principles for the Hankel transform by means of the Hankel multiplier. Then we define the wavelet Hankel multiplier and study its boundedness and Schatten-class properties. Finally, we prove that the wavelet Hankel multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator, for which we deduce a trace formula.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 1 (2018), 144-166.

Dates
Received: 24 November 2016
Accepted: 12 March 2017
First available in Project Euclid: 8 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1510110961

Digital Object Identifier
doi:10.1215/17358787-2017-0051

Mathematical Reviews number (MathSciNet)
MR3745578

Zentralblatt MATH identifier
1383.81120

Subjects
Primary: 81S30: Phase-space methods including Wigner distributions, etc.
Secondary: 94A12: Signal theory (characterization, reconstruction, filtering, etc.) 45P05: Integral operators [See also 47B38, 47G10] 42C25: Uniqueness and localization for orthogonal series 42C40: Wavelets and other special systems

Keywords
Hankel multiplier localization operator uncertainty principle Nash inequality Carlson inequality

Citation

Ghobber, Saifallah. A variant of the Hankel multiplier. Banach J. Math. Anal. 12 (2018), no. 1, 144--166. doi:10.1215/17358787-2017-0051. https://projecteuclid.org/euclid.bjma/1510110961


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References

  • [1] L. D. Abreu and J. M. Pereira,Measures of localization and quantitative Nyquist densities, Appl. Comput. Harmon. Anal.38(2015), no. 3, 524–534.
  • [2] P. Boggiatto, E. Carypis, and A. Oliaro,Two aspects of the Donoho–Stark uncertainty principle, J. Math. Anal. Appl.434(2016), no. 2, 1489–1503.
  • [3] P. C. Bowie,Uncertainty inequalities for Hankel transforms, SIAM J. Math. Anal.2(1971), 601–606.
  • [4] I. Daubechies,Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inform. Theory34(1988), no. 4, 605–612.
  • [5] D. L. Donoho and P. B. Stark,Uncertainty principles and signal recovery, SIAM J. Appl. Math.49(1989), no. 3, 906–931.
  • [6] S. Ghobber,Uncertainty principles involving $L^{1}$-norms for the Dunkl transform, Integral Transforms Spec. Funct.24(2013), no. 6, 491–501.
  • [7] S. Ghobber,Variations on uncertainty principles for integral operators, Appl. Anal.93(2014), no. 5, 1057–1072.
  • [8] S. Ghobber,Phase space localization of orthonormal sequences in $L^{2}_{\alpha}(\mathbb{R}_{+})$, J. Approx. Theory189(2015), 123–136.
  • [9] S. Ghobber and P. Jaming,Strong annihilating pairs for the Fourier–Bessel transform, J. Math. Anal. Appl.377(2011), no. 2, 501–515.
  • [10] S. Ghobber and P. Jaming,The Logvinenko–Sereda theorem for the Fourier–Bessel transform, Integral Transforms Spec. Funct.24(2013), no. 6, 470–484.
  • [11] S. Ghobber and P. Jaming,Uncertainty principles for integral operators, Studia Math.220(2014), 197–220.
  • [12] L. Gosse,A Donoho–Stark criterion for stable signal recovery in discrete wavelet subspaces, J. Comput. Appl. Math.235(2011), no. 17, 5024–5039.
  • [13] V. Havin and B. Jöricke,The Uncertainty Principle in Harmonic Analysis, Ergeb. Math. Grenzgeb (3)28, Springer, Berlin, 1994.
  • [14] H. J. Landau,On Szegö’s eigenvalue distribution theorem and non-Hermitian kernels, J. Anal. Math.28(1975), 335–357.
  • [15] H. J. Landau and H. O. Pollak,Prolate spheroidal wave functions, Fourier analysis and uncertainty, II, Bell Syst. Tech. J.40(1961), 65–84.
  • [16] H. J. Landau and H. O. Pollak,Prolate spheroidal wave functions, Fourier analysis and uncertainty, III: The dimension of the space of essentially time- and band-limited signals, Bell Syst. Tech. J.41(1962), 1295–1336.
  • [17] S. Omri,Local uncertainty principle for the Hankel transform, Integral Transforms Spec. Funct.21(2010), no. 9-10, 703–712.
  • [18] M. Reed and B. Simon,Methods of Modern Mathematical Physics, I: Functional Analysis, 2nd ed., Academic Press, New York, 1980.
  • [19] M. Rösler and M. Voit,An uncertainty principle for Hankel transforms, Proc. Amer. Math. Soc.127(1999), no. 1, 183–194.
  • [20] D. Slepian,Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev.25(1983), no. 3, 379–393.
  • [21] D. Slepian and H. O. Pollak,Prolate spheroidal wave functions, Fourier analysis and uncertainty, I, Bell Syst. Tech. J.40(1961), 43–63.
  • [22] E. M. Stein,Interpolation of linear operators, Trans. Amer. Math. Soc.83(1956), 482–492.
  • [23] V. K. Tuan,Uncertainty principles for the Hankel transform, Integral Transforms Spec. Funct.18(2007), no. 5–6, 369–381.
  • [24] G. N. Watson,A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1944.
  • [25] M. W. Wong,Wavelet Transforms and Localization Operators, Oper. Theory Adv. Appl.136, Birkhäuser, Basel, 2002.