Opuscula Math. 35, no. 6 (2015), 867-887
http://dx.doi.org/10.7494/OpMath.2015.35.6.867

 
Opuscula Mathematica

Inversion of the Riemann-Liouville operator and its dual using wavelets

C. Baccar
N. B. Hamadi
H. Herch
F. Meherzi

Abstract. We define and study the generalized continuous wavelet transform associated with the Riemann-Liouville operator that we use to express the new inversion formulas of the Riemann-Liouville operator and its dual.

Keywords: inverse problem, Riemann-Liouville operator, Fourier transform, wavelets.

Mathematics Subject Classification: 35R30, 42B10, 42C40.

Full text (pdf)

  1. M. Agranovsky, P. Kuchment, E.T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal. 248 (2007) 2, 344-386.
  2. L.E. Andersson, On the determination of a function from spherical averages, SIAM J. Math. Anal. 19 (1988) 1, 214-234.
  3. L.E. Andersson, H. Helesten, An inverse method for the processing of synthetic aperture radar data, Inverse Problems 3 (1987) 1, 111-124.
  4. C. Baccar, N.B. Hamadi, L.T. Rachdi, Inversion formulas for the Riemann-Liouville transform and its dual associated with singular partial differential operators, Int. J. Math. Math. Sci. 2006 (2006) 86238, 1-26.
  5. C. Baccar, N.B. Hamadi, L.T. Rachdi, Best approximation for Weierstrass transform connected with Riemann-Liouville operator, Commun. Math. Anal. 5 (2008) 1, 65-83.
  6. C. Baccar, L.T. Rachdi, Spaces of \(D_{L^p}\)-type and a convolution product associated with the Riemann-Liouville operator, Bull. Math. Anal. Appl. 1 (2009) 3, 16-41.
  7. J. Cohen, N. Bleistein, Velocity inversion procedure for acoustic waves, Geophysics 44 (1979) 6, 1077-1087.
  8. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990) 5, 961-1005.
  9. A. Erdely, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions II, McGraw-Hill, New York, 1953.
  10. J.A. Fawcett, Inversion of N-dimensional spherical averages, SIAM J. Appl. Math. 45 (1985) 2, 336-341.
  11. E. Foufoula-Georgiou, P. Kumar, Wavelets in Geophysics, vol. 4 of Wavelet Analysis and Its Applications, Academic Press, New York, 1994.
  12. A. Grossman, J. Morlet, T. Paul, Transforms associated to square integrable group representations I. General results, J. Math. Phys. 26 (1985), 2473-2479.
  13. A. Grossman, J. Morlet, T. Paul, Transforms associated to square integrable group representations II. Examples, Ann. Henri Poincaré 45 (1986), 293-309.
  14. N.B. Hamadi, L.T. Rachdi, Weyl transforms associated with the Riemann-Liouville operator, Int. J. Math. Sci. 2006 (2006), 94768, 1-19.
  15. S. Helgason, The Radon Transform, Birkhäuser, Boston, 1999.
  16. M. Herberthson, A numerical implementation of an inverse formula for CARABAS Raw Data, National Defense Research Institute, Internal Report, D 30430-3.2, FOA, Linköping, Sweden, 1986.
  17. W. Joines, Y. Zhang, C. Li, R. Jirtle, The measured electrical properties of normal and malignant human tissue from 50 to 900 Mhz, Med. Phys. 21 (1994) 4, 547-550.
  18. T.H. Koornwinder, The continuous wavelet transform, [in:] Wavelets: An Elementary Treatment of Theory and Applications, vol. 1 of Series in Approximations and Decompositions, World Scientific, Singapore, 1993, 27-48.
  19. R.S. Laugesen, N. Weaver, G.L. Weiss, E.N. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal. 12 (2002) 1, 89-102.
  20. N.N. Lebedev, Special Functions and Their Applications, Courier Dover Publications, New York, 1972.
  21. D. Ludwig, The Radon transform on euclidean space, Comm. Pure Appl. Math. 19 (1966) 1, 49-81.
  22. Y. Meyer, Wavelets and Operators, vol. 1 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.
  23. M.M. Nessibi, L.T. Rachdi, K. Trimèche, Ranges and inversion formulas for spherical mean operator and its dual, J. Math. Anal. Appl. 196 (1995) 3, 861-884.
  24. M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 2, 337-363.
  25. S. Omri, L.T. Rachdi, Heisenberg-Pauli-Weyl uncertainty principle for the Riemann-Liouville operator, J. Inequal. Pure Appl. Math. 9 (2008) 3, Art. 88, 23 pp.
  26. D.C. Solmon, Asymptotic formulas for the dual Radon transform, Math. Z. 195 (1987) 3, 321-343.
  27. E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956) 2, 482-492.
  28. K. Trimèche, Weyl integral transform and Paley-Wiener theorem associated with a singular differential operator on \((0,+\infty)\), J. Math. Pures Appl. 60 (1981), 51-98 [in French].
  29. K. Trimèche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comput. Harmon. Anal. 4 (1997) 1, 97-112.
  30. G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995.
  • C. Baccar
  • Higher Institute of Informatics of El Manar 2, Department of Applied Mathematics, Rue Abou Raïhan El Bayrouni - 2080 Ariana, Tunisia
  • N. B. Hamadi
  • Department of Mathematics, Preparatory Institute for Engineering Studies El Manar, 2092 El Manar 2 Tunis, Tunisia
  • H. Herch
  • F. Meherzi
  • Communicated by Semyon B. Yakubovich.
  • Received: 2014-11-10.
  • Revised: 2014-12-29.
  • Accepted: 2015-01-05.
  • Published online: 2015-06-06.
Opuscula Mathematica - cover

Cite this article as:
C. Baccar, N. B. Hamadi, H. Herch, F. Meherzi, Inversion of the Riemann-Liouville operator and its dual using wavelets, Opuscula Math. 35, no. 6 (2015), 867-887, http://dx.doi.org/10.7494/OpMath.2015.35.6.867

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.