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Gabor frame decomposition of evolution operators and applications

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Abstract

We compute the Gabor matrix for Schrödinger-type evolution operators. Precisely, we analyze the heat equation, already presented in Cordero et al. (Gabor representations of evolution operators, 2012), giving the exact expression of the Gabor matrix which leads to better numerical evaluations. Then, using asymptotic integration techniques, we obtain an upper bound for the Gabor matrix in one-dimension for the generalized heat equation, new in the literature. Using Maple software, we show numeric representations of the coefficients’ decay. Finally, we show the super-exponential decay of the coefficients of the Gabor matrix for the harmonic repulsor, together with some numerical evaluations. This work is the natural prosecution of the ideas presented in Cordero et al. (Gabor representations of evolution operators, 2012) and Cordero et al. (Sparsity of Gabor representation of Schrödinger propagators, Appl Comput Harmonic Appl 26:357–370, 2009).

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References

  1. Christensen, O., Feichtinger, H.G., Paukner, S.: Gabor analysis for imaging. In: Handbook of Mathematical Methods in Imaging, pp. 1271–1307. Springer, Berlin (2011)

  2. Cordero, E., Nicola, F., Rodino, L.: Gabor representations of evolution operators (2012). ArXiv e-prints

  3. Cordero, E., Gröchenig, K., Nicola, F., Rodino, L.: Wiener algebras of Fourier integral operators. J. Math. Pures Appl. (9) 99(2), 219–233 (2013)

    Google Scholar 

  4. Cordero, Elena, Nicola, Fabio, Rodino, Luigi: Sparsity of Gabor representation of Schrödinger propagators. Appl. Comput. Harmonic Anal. 26(3), 357–370 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cordero, Elena, Nicola, Fabio, Rodino, Luigi: Time–frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal. 9(1), 1–21 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cordero, E., Nicola, F., Rodino, L.: Time–frequency analysis of schroedinger propagators. In: Evolution Equations of Hyperbolic and Schrödinger Type, pp. 63–85. Springer, Berlin (2012)

  7. Eastham, M.S.P.: Asymptotic formulae of Liouville–Green type for higher-order differential equations. J. Lond. Math. Soc. (2) 28(3), 507–518 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. Folland, G.B.: Harmonic Analysis in Phase Space, volume 122 of Annals of Mathematics Studies. Princeton University Press, Princeton (1989)

    Google Scholar 

  9. Gelfand, I.M., Shilov, G.E.: Generalized Functions, vol. 2–3. Academic Press, New York (1967)

  10. Grafakos, Loukas, Sansing, Christopher: Gabor frames and directional time–frequency analysis. Appl. Comput. Harmonic Anal. 25(1), 47–67 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gröchenig, Karlheinz: Foundations of Time–Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001)

    Google Scholar 

  12. Gröchenig, Karlheinz: Time–frequency analysis of Sjöstrand’s class. Rev. Mat. Iberoam. 22(2), 703–724 (2006)

  13. Heil, C.: A Basis Theory Primer. Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2011)

    Book  Google Scholar 

  14. Hinton, D.B.: Asymptotic behavior of solutions of \((ry^{(m)})^{(k)}\pm qy=0\). J. Diff. Equ. 4, 590–596 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lamoureux, M.P., Margrave, G.F.: An introduction to numerical methods of pseudodifferential operators. In: Pseudo-Differential Operators, volume 1949 of Lecture Notes in Mathematics, pp. 79–133. Springer, Berlin, (2008)

  16. Leach, P.G.L.: Sl(3, R) and the repulsive oscillator. J. Phys. A 13(6), 1991–2000 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces, volume 4 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel (2010)

    Book  Google Scholar 

  18. Onchis, D.M., Real, P., Gillich, G.-R.: Gabor frames and topology-based strategies for astronomical images. (2010)

  19. Seip, K.: Density theorems for sampling and interpolation in the Bargmann–Fock space I. J. Reine Angew. Math. 429, 91–106 (1992)

    MATH  MathSciNet  Google Scholar 

  20. Seip, K., Wallstén, R.: Density theorems for sampling and interpolation in the Bargmann–Fock space II. J. Reine Angew. Math. 429, 107–113 (1992)

    MATH  MathSciNet  Google Scholar 

  21. Strohmer, T.: Pseudodifferential operators and banach algebras in mobile communications. Appl. Comput. Harmonic Anal. 20(2), 237–249 (2006). Computational harmonic analysis-Part 3

    Article  MATH  MathSciNet  Google Scholar 

  22. ter Morsche, H., Oonincx, P.J.: On the integral representations for metaplectic operators. J. Fourier Anal. Appl. 8(3), 245–257 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Dover, New York (1987). Reprint of the 1976 edition

    MATH  Google Scholar 

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Acknowledgments

I am sincerely grateful to Professor E. Cordero for the fruitful discussion, valuable advices, constructive criticism and constant review of this work. I would like to thank Professors E. Cordero and L. Rodino for inspiring this paper. I also wish to thank M. Borsero for the useful suggestions, and the final review aimed at improving the readability of the paper. Finally, I am thankful for the enormous work of the anonymous reviewer who suggested important corrections that gave consistency to the paper.

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  1. Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 , Turin, TO, Italy

    Michele Berra

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  1. Michele Berra
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Correspondence to Michele Berra.

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Berra, M. Gabor frame decomposition of evolution operators and applications. J. Pseudo-Differ. Oper. Appl. 5, 277–304 (2014). https://doi.org/10.1007/s11868-014-0090-8

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