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Results in Mathematics

, Volume 63, Issue 1–2, pp 511–528 | Cite as

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

  • M. M. Rodrigues
  • N. VieiraEmail author
  • S. Yakubovich
Article

Abstract

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L 2.

Mathematics Subject Classification (2000)

44A15 44A05 44A35 33C10 45A05 

Keywords

Kontorovich–Lebedev transform Generalized Mehler–Fock transform Modified Bessel function associated Legendre functions convolution integral equations index integrals 

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References

  1. 1.
    Erdelyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher Transcendental Functions, vol. 1–2. McGraw-Hill, New York (1953)Google Scholar
  2. 2.
    Ferrel T.L.: Modulation of collective electronic effects in foils by the electron tunneling microscope. Nucl. Instrum. Methods Phys. Res. B 96, 483–485 (1995)CrossRefGoogle Scholar
  3. 3.
    Lebedev N.N.: Special Functions and their Applications. Prentice-Hall Inc., Englewood Cliffs (1965)zbMATHGoogle Scholar
  4. 4.
    Nasim C.: The Mehler–Fock transform of general order and arbitrary index and its inversion. Int. J. Math. Math. Sci. 7(1), 171–180 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 1: Elementary Functions. Gordon and Breach, New York (1986)Google Scholar
  6. 6.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 2: Special Functions. Gordon and Breach, New York (1986)Google Scholar
  7. 7.
    Prudnikov, A.P., Brychkov Yu.A., Marichev, O.I.: Integrals and Series, vol. 3: More Special Functions. Gordon and Breach, New York (1989)Google Scholar
  8. 8.
    Sneddon I.N.: The Uses of Integrals Transforms. McGraw-Hill, New York (1972)Google Scholar
  9. 9.
    Yakubovich, S.: An index integral and convolution operator related to the Kontorovich–Lebedev and Mehler–Fock transforms. Complex. Anal. Oper. Theory. doi: 10.1007/s11785-010-0112-3
  10. 10.
    Passian A., Koucheckian S., Yakubovich S.: Index integral representations for connection between cartesian, cylindrical, and spheroidal systems. Integral Transforms Spec. Funct 22(8), 549–560 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Vilenkin N.Ja.: The matrix elements of irreducible unitary representations of a group of Lobachevsky space motions and the generalized Fock–Mehler transformations. Dokl. Akad. Nauk SSSR 118, 219–222 (1958) (in Russian)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yakubovich S.: On the least values of L p-norms for the Kontorovich–Lebedev transform and its convolution. J. Approx. Theory 131, 231–242 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Yakubovich S., Saigo M.: On the Mehler–Fock transform in L p-space. Math. Nachr. 185, 261–277 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Yakubovich S.: Index Transforms. World Scientific Publishing Company, Singapore (1996)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  2. 2.Department of Mathematics, Center of Mathematics of University of Porto, Faculty of ScienceUniversity of PortoPortoPortugal
  3. 3.Department of Mathematics, Faculty of ScienceUniversity of PortoPortoPortugal

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