Let G be a locally compact Lie group and its Lie algebra. We consider a fuzzy analogue of G, denoted by called a fuzzy Lie group. Spherical functions on are constructed and a version of the existence result of the Helgason-spherical function on G is then established on .
References
[1]
Zadeh, L.A. (1965) Fuzzy Sets. Fuzzy Information and Control, 8, 177-200. https://doi.org/10.1016/S0019-9958(65)90241-X
[2]
Kastaras, A.K. (1981) Topological Linear Spaces I. Fuzzy Sets and Systems, 6, 85-95. https://doi.org/10.1016/0165-0114(81)90082-8
[3]
Kastaras, A.K. (1984) Topological Linear Spaces II. Fuzzy Sets and Systems, 12, 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
[4]
Kastaras, A.K. and Liu, D.B. (1977) Fuzzy Vector Spaces and Fuzzy Topological Vector Spaces. Journal of Mathematical Analysis and Applications, 58, 135-146. https://doi.org/10.1016/0022-247X(77)90233-5
[5]
Kubiak, T. (1985) On Fuzzy Topologies. Ph. D. Thesis, Adam Mickiewicz, Poznan.
[6]
Sostak, A.P. (1985) On a Fuzzy Topological Structure. Proceedings of the 13th Winter School on Abstract Analysis, Circolo Matematico di Palermo, 89-103.
[7]
Egwe, M.E. and Oyewo, R.A. (2021) A Fixed Point Theorem on Fuzzy Locally Convex Spaces. Annals of Fuzzy Mathematics and Informatics, 22, 17-24. https://doi.org/10.20944/preprints202206.0311.v1
[8]
Chakraborty, M. and Ahsanullah, T. (1992) Fuzzy Topology on Fuzzy Sets and Tolerance Topology. Fuzzy Sets and Systems, 45, 103-108. https://doi.org/10.1016/0165-0114(92)90096-M
[9]
Akram, M. (2018) Fuzzy Lie Algebras. Springer Verlag, Berlin. https://doi.org/10.1007/978-981-13-3221-0
[10]
Nadjafikhah, M. and Bakhshandeh-Chamazkoti, R. (2009) Fuzzy Lie Groups. Mathematical Sciences, arXiv: 0908.0254. https://arxiv.org/abs/0908.0254v1
[11]
Godement, R. (1952) A Theory of Spherical Functions I.Trans. Transactions of the American Mathematical Society, 73, 496-556. https://doi.org/10.1090/S0002-9947-1952-0052444-2
[12]
Chandra, H. (1958) Spherical Functions on a Semisimple Lie Group, I. American Mathematical Society, 80, 241-310. https://doi.org/10.2307/2372786
[13]
Egwe, M.E. (2020) On Fixed Point Theorem in Non-Archimedean Fuzzy Normed Spaces. International Journal of Analysis and Applications, 18, 99-103.
[14]
Yeol, J.C., Rassias, T.M. and Saadati, R. (2018) Fuzzy Operator Theory in Mathematical Analysis. Springer Verlag, Berlin.
[15]
Helgason, S. (1984) Groups and Geometric Analysis: Radon Transforms, Invariant Differential Operators and Spherical Functions: Volume 1. Academic Press, Cambridge.
[16]
Egwe, M.E. and Bassey, U.N. (2013) On Isomorphism between Certain Group Algebras on the Heisenberg Group. Journal of Mathematical Physics, Analysis, Geometry, 9, 150-164.
[17]
Hilgert, J. and Olafsson, G. (1997) Causal Symmetric Spaces: Geometry and Harmonic Analysis.Academic Press Inc., Cambridge.
its Lie algebra. We consider a fuzzy analogue of G, denoted by 
called a fuzzy Lie group. Spherical functions on 
are constructed and a version of the existence result of the Helgason-spherical function on G is then established on 
.
