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 Volume 8, Issue 11 (November 2021), Pages: 1-9


 Original Research Paper

 Title: Analysis of Gegenbauer kernel filtration on the hypersphere

 Author(s): Louis Omenyi 1, *, McSylvester Omaba 2, Emmanuel Nwaeze 1, Michael Uchenna 1


 1Department of Mathematics and Statistics, Alex Ekwueme Federal University, Ndufu-Alike, Nigeria
 2Department of Mathematics, College of Science, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia

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In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated. 

 © 2021 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (

 Keywords: Spherical Laplacian, Hypersphere, Gegenbauer kernel, Filtration, Cauchy wave problem

 Article History: Received 28 May 2021, Received in revised form 10 August 2021, Accepted 25 August 2021


The first author thanks the leadership and members of the Seminar and Research Committee of the Department of Mathematics and Statistics of the Alex Ekwueme Federal University, Ndufu-Alike for their valuable scientific discussions that facilitated the completion of this research. All the authors express sincere gratitude to their family members for providing the support to work.

 Compliance with ethical standards

 Conflict of interest: The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.


 Omenyi L, Omaba M, and Nwaeze E et al. (2021). Analysis of Gegenbauer kernel filtration on the hypersphere. International Journal of Advanced and Applied Sciences, 8(11): 1-9

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 References (24)

  1. Antoine JP and Vandergheynst P (1999). Wavelets on the 2-sphere: A group-theoretical approach. Applied and Computational Harmonic Analysis, 7(3): 262-291.   [Google Scholar]
  2. Assche WV, Yánez RJ, González-Férez R, and Dehesa JS (2000). Functionals of Gegenbauer polynomials and d-dimensional hydrogenic momentum expectation values. Journal of Mathematical Physics, 41(9): 6600-6613.   [Google Scholar]
  3. Atkinson K and Han W (2012). Spherical harmonics and approximations on the unit sphere: An introduction. Volume 2044, Springer Science and Business Media, Berlin, Germany.   [Google Scholar]
  4. Aubin T (1998). Some nonlinear problems in Riemannian geometry. Springer, Berlin, Germany.   [Google Scholar]
  5. Avery JS (2012). Hyperspherical harmonics: Applications in quantum theory. Volume 5, Springer Science and Business Media, Berlin, Germany.   [Google Scholar]
  6. Bezubik A and Strasburger A (2006). A new form of the spherical expansion of zonal functions and Fourier transforms of SO (d)-finite functions. Symmetry, Integrability and Geometry: Methods and Applications, 2: 033.   [Google Scholar]
  7. Bogdanova I, Vandergheynst P, Antoine JP, Jacques L, and Morvidone M (2005). Stereographic wavelet frames on the sphere. Applied and Computational Harmonic Analysis, 19(2): 223-252.   [Google Scholar]
  8. Bulow T (2004). Spherical diffusion for 3D surface smoothing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(12): 1650-1654.   [Google Scholar] PMid:15573826
  9. Camporesi R (1990). Harmonic analysis and propagators on homogeneous spaces. Physics Reports, 196(1-2): 1-134.   [Google Scholar]
  10. Claessens SJ (2016). Spherical harmonic analysis of a harmonic function given on a spheroid. Geophysical Journal International, 206(1): 142-151.   [Google Scholar]
  11. Cohl HS and Palmer RM (2015). Fourier and Gegenbauer expansions for a fundamental solution of Laplace's equation in hyperspherical geometry. Symmetry, Integrability and Geometry: Methods and Applications, 11: 015.   [Google Scholar]
  12. Dai F and Xu Y (2013). Convolution operator and spherical harmonic expansion. In: Dai F and Xu Y (Eds.), Approximation theory and harmonic analysis on spheres and balls: 29-51. Springer, New York, USA.   [Google Scholar] PMCid:PMC3569964
  13. Drake JB, Worley P, and D’Azevedo E (2008). Spherical harmonic transform algorithms. ACM Transactions on Mathematical Software, 35(3): 111-131.   [Google Scholar]
  14. Driscoll JR and Healy DM (1994). Computing Fourier transforms and convolutions on the 2-sphere. Advances in Applied Mathematics, 15(2): 202-250.   [Google Scholar]
  15. Healy DM, Rockmore DN, Kostelec PJ, and Moore S (2003). FFTs for the 2-sphere-improvements and variations. Journal of Fourier Analysis and Applications, 9(4): 341-385.   [Google Scholar]
  16. Jost J and Jost J (2008). Riemannian geometry and geometric analysis. Volume 42005, Springer, Berlin, Germany.   [Google Scholar]
  17. Lee JM (2003). Introduction to smooth manifolds: Graduate texts in mathematics. Springer Science and Business Media, New York, USA.   [Google Scholar]
  18. Morimoto M (1998). Analytic functionals on the sphere. Translations of Mathematical Monographs, American Mathematical Society, Providence, USA.   [Google Scholar]
  19. Omenyi L (2014). On the second variation of the spectral zeta function of the Laplacian on homogeneous Riemanniann manifolds. Ph.D. Dissertation, Loughborough University, Loughborough, UK.   [Google Scholar]
  20. Omenyi L and Uchenna M (2019). Global analysis on Riemannian manifold. The Australian Journal of Mathematical Analysis and Applications, 16(2): 1-17.   [Google Scholar]
  21. Pinchover Y and Rubinstein J (2005). An introduction to partial differential equations. Volume 10, Cambridge University Press, Cambridge, USA.   [Google Scholar]
  22. Strasburger A (1993). A generalization of the Bochner identity. Expositiones Mathematicae, 11: 153-157.   [Google Scholar]
  23. Szekeres P (2004). A course in modern mathematical physics: Groups, Hilbert space and differential geometry. Cambridge University Press, Cambridge, UK.   [Google Scholar]
  24. Wong MW (2006). Weyl transforms, heat kernels, green functions and Riemann zeta functions on compact lie groups. In: Toft J, Wong MW, and Zhu H (Eds.), Modern trends in Pseudo-differential operators: 67-85. Birkhäuser, Basel, Switzerland.   [Google Scholar]