Positive Definite Kernels and Radial Distributions on the Euclidean Motion Group

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-55
Published: Mar 3, 2025
Cite as:
U.E. Edeke, J.A. Odey, E.E. Essien, A.O. Otiko, D.O. Egete, J.N. Ezeorah, E.E. Bassey, B.I. Ele, Positive Definite Kernels and Radial Distributions on the Euclidean Motion Group, Int. J. Anal. Appl., 23 (2025), 55.

Main Article Content

U.E. Edeke, J.A. Odey, E.E. Essien, A.O. Otiko, D.O. Egete, J.N. Ezeorah, E.E. Bassey, B.I. Ele

Abstract

Let G = R2⋊T be the Euclidean motion group and let K(λ,t) = I0(λ)δ(t) be a distribution on G, where I0(λ) is the Bessel function of order zero and δ(t) is the Dirac measure on SO(2)≅T, the circle group. In this work, it is proved, among other things, that the distribution K(λ,t) is tempered, positive definite, bounded and radial. Furthermore, a description of Levy-Schoenberg Kernels on the homogenous space of SE(2) is presented.

Article Details

References

  1. F. Astengo, B. Di Blasio, F. Ricci, On the Schwartz Correspondence for Gelfand Pairs of Polynomial Growth, Rendiconti Lincei, Mat. Appl. 32 (2021), 79–96. https://doi.org/10.4171/rlm/927.
  2. A.O. Barut, R. Raczka, Theory of Group Representations and Applications, Polish Scientific Publishers, 1980.
  3. U.N. Bassey, U.E. Edeke, The Radial Part of an Invariant Differential Operator on the Euclidean Motion Groups, J. Nigerian Math. Soc. 43 (2024), 237-251.
  4. U.N. Bassey, U.E. Edeke, Convolution Equation and Operators on the Euclidean Motion Group, J. Nigerian Soc. Phys. Sci. 6 (2024), 2029. https://doi.org/10.46481/jnsps.2024.2029.
  5. G.S. Chirikjian, Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups, CRC Press, 2000. https://doi.org/10.1201/9781420041767.
  6. U.E. Edeke, R.D. Ariyo, O.C. Dada, A Class of Positive Definite Spherical Functions on the Euclidean Motion Groups, Glob. J. Pure Appl. Sci. 30 (2024), 551–558. https://doi.org/10.4314/gjpas.v30i4.13.
  7. M.E. Egwe, A Radial Distribution on the Heisenberg Group, Pure Math. Sci. 3 (2014), 105–112. https://doi.org/10.12988/pms.2014.436.
  8. F.F. Ruffino, The Topology of the Spectrum for Gelfand Pairs on Lie Groups, Boll. Un. Mat. Ital. 10-B (2007), 569-579. http://eudml.org/doc/290444.
  9. J. Kim, M.W. Wong, Positive Definite Temperature Functions on the Heisenberg Group, Appl. Anal. 85 (2006), 987–1000. https://doi.org/10.1080/00036810600808901.
  10. M. Sugiura, Unitary Representations and Harmonic Analysis: An Introduction, North-Holland, Amsterdam, 1990.
  11. N.Ja. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions, Kluwer Academic Publishers, 1991.
  12. C.E. Yarman, B. Yazici, A Wiener Filtering Approach over the Euclidean Motion Group for Radon Transform Inversion, in: M. Sonka, J.M. Fitzpatrick (Eds.), San Diego, CA, 2003: p. 1884. https://doi.org/10.1117/12.481372.
  13. R. Gangolli, Positive Definite Kernels on Homogeneous Spaces and Certain Stochastic Processes Related to Levy’s Brownian Motion of Several Parameters, Ann. Inst. H. Poincaré Probab. Stat. 3 (1967), 121-226.