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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.13 no.3 Temuco oct. 2011

http://dx.doi.org/10.4067/S0719-06462011000300006 

CUBO A Mathematical Journal Vol.13, NQ03, (91-115). October 2011

 

Uncertainty principle for the Riemann-Liouville operator

 

Hleili Khaled, Omri Slim and Lakhdar T. Rachdi

Faculty of Applied Mathematics,Departement de Mathematiques et d'Informatique, Institut national des sciences appliquees et de Thechnologie, Centre Urbain Nord BP 676 - 1080 Tunis cedex, Tunisia, email: khaled.hleili@gmail.com

Departement de Mathematiques Appliquees, Institut preparatoire aux (etudes d'ingenieurs, Campus universitaire Mrezka - 8000 Nabeul, Tunisia. email: slim.omri@issig.rnu.tn

Departement de Math ematiques, Faculte des Sciences de Tunis, 2092 El Manar II, Tunisia. email: lahhdartannech.rachdi@fst.rnu.tn


ABSTRACT

A Beurling-Hormander theorem's is proved for the Fourier transform connected with the Riemann-Liouville operator. Nextly, Gelfand-Shilov and Cowling-Price type theorems are established.

Keywords: Beurling-Hormander theorem, Gelfand-Shilov theorem, Cowling- Price theorem, Fourier transform, Riemann-Liouville operator.

Mathematics Subject Classification: 43A32; 42B10.


RESUMEN

Se demuestra el teorema de Beurling-Hormander por la transformada de Fourier conectada con el operador de Riemann-Liouville. Ademas, se establecen teoremas tipo de Gelfand-Shilov y Cowling-Price.


References

[1] G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, New-York 1999.         [ Links ]

[2] C. Baccar, N. B. Hamadi and 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), pp 1-26.         [ Links ]

[3] S. Ben Farah and K. Mokni, Uncertainty Principle and the (Lp, Lq) version of Morgans theorem on some groups, Russ. J. Math. Phys., 10 No. 3 (2003), pp 245-260.         [ Links ]

[4] A. Beurling, The collected works of Arne Beurling, Birkhuser., Vol.1-2, Boston 1989.         [ Links ]

[5] A. Bonami, B. Demange, and P. Jaming, Hermite functions and uncertainty priciples for the Fourier and the widowed Fourier transforms, Rev. Mat. Iberoamericana., 19 (2003), pp 23-55.         [ Links ]

[6] L. Bouattour and K. Trimeche, An analogue of the Beurling-Hormander's theorem for the Chebli-Trimeche transform, Glob. J. Pure Appl. Math., 1 No. 3 (2005), pp 342-357.         [ Links ]

[7] B. Chabat, Introduction a l'analyse complexe, Edition Mir., Vol.2, Moscou 1985.         [ Links ]

[8] M.G. Cowling and J. F. Price, Generalizations of Heisenbergs inequality in Harmonic analysis, (Cortona, 1982), Lecture Notes in Math., 992 (1983), pp 443-449.         [ Links ]

[9] A. Erdely and all, Asymptotic expansions, Dover publications, New-York 1956.         [ Links ]

[10] A. Erdely and all, Tables of integral transforms, Mc Graw-Hill Book Compagny., Vol.2, New York 1954.         [ Links ]

[11] G. B. Folland, Real Analysis Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley and Sons., New York 1984.         [ Links ]

[12] G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), pp 207-238.         [ Links ]

[13] I.M. Gelfand and G.E. Shilov, Fourier transforms of rapidly increasing functions and questions of uniqueness of the solution of Cauchy's problem, Uspekhi Mat. Nauk., 8 (1953), pp 3-54.         [ Links ]

[14] G. H. Hardy, A theorem concerning Fourier transform, J. London. Math. Soc., 8 (1933), pp 227-231.         [ Links ]

[15] V. Havin and B. Joricke, An uncertainty principle in harmonic analysis, Springer Verlag., Berlin 1994.         [ Links ]

[16] L. Hormander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat., 29 (1991), pp 237-240.         [ Links ]

[17] L. Kamoun and K. Trimeche, An Analogue of Beurling-Hormander's Theorem Associated with Partial Differential Operators, Meditter. J. Math., 2 (2005), pp 243-258.         [ Links ]

[18] N. N. Lebedev, Special Functions and their applications, Dover publications., New-York 1972.         [ Links ]

[19] G. W. Morgan, A note on Fourier transforms, J. London. Math. Soc., 9 (1934), pp 178-192.         [ Links ]

[20] S. Omri and L. T. Rachdi, An Lp — Lq version of Morgan's theorem associated with Riemann-Liouville transform, Int. J. Math. Anal., 1 No. 17 (2007), pp 805-824.         [ Links ]

[21] S. Omri and L. T. Rachdi, Heisenberg-Pauli-Weyl uncertainty principle for the Riemann-Liouville Operator, J. Ineq. Pure and Appl. Math., 9 (2008), Iss. 3, Art 88.         [ Links ]

[22] K. Trimeche, Beurling-Hormander's theorem for the Dunkl transform, Glob. J. Pure Appl. Math., 2 No. 3 (2006), pp 181-196.         [ Links ]

[23] K. Trimeche, Inversion of the Lions translation operator using genaralized wavelets, Appl. Comput. Harmonic Anal., 4 (1997), pp 97-112.         [ Links ]

[24] K. Trimeche, Transformation integrale de Weyl et theoreme de Paley-Wienner associes a un operateur differentiel singulier sur (0, +oo), J. Math. Pures Appl., 60 (1981), pp 51-98.         [ Links ]

[25] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge univ. Press., 2nd ed, Cambridge 1959.         [ Links ]

[26] S.B. Yakubovich, Uncertainty principles for the Kontorovich-Lebedev transform, Math. Mod- ell. Anal., 13 No. 2 (2008), pp 289-302.         [ Links ]


Received: July 2010. Revised: August 2010.