Boehmians on the Sphere
by
Mitsuo Morimoto
Department of Mathematics, International Christian University , Osawa 3-10-2, Mitaka, Tokyo, 181-8585 Japan

The Boehmians were first introduced by J.Mikusi\'nski and P.Mikusi\'nski in 1981 as a generalization of regular (Mikusi\'nski) operators considered by T.K.Boehme in 1973. Although the definition of Boehmians was given in an abstract algebraic form, it was mainly applied to generalized functions on the Euclidean space. D.Nemzer considered Boehmians on the unit circle in 1989, 1990, and 1991 by identifying them with periodic Boehmians on the real line.

In [1] and [2] the Boehmians were constructed on the sphere. Let d >= 2 be an integer. We denote by S^d={x=(x₁, x₂, ... x_d+1 ): x₁²+x₂² ... x_d+1²=1 } the d-dimensional sphere in R^d+1 and by G=SO(d+1) the group of rotations. The group G=SO(d+1) is a non-commutative compact Lie group. Therefore, the convolution algebra C(G) of continuous functions on G is not commutative. The group G acts on S^d transitively. The isotropy subgroup of a point x₀ in S^d, H={T in G : Tx₀=x₀}, is isomorphic to SO(d) and we have S^d=X/H. Functions on S^d can be considered as left H-invariant functions on G. Using this group structure Boehmians on S^d were constructed and their spherical harmonic expansions were studied. The methods employed in [1] and [2] looked different. We shall discuss here their relation.

References

[1] Piotr Miku\'sinski and Mitsuo Morimoto: Boehmians on the sphere and their spherical harmonic expansions, Fractional Calculus and Applied Analysis, 4(2001), 25-35.

[2] P.Miku\'sinski, B.A.Pyle: Boehmians on the sphere, Integral Transforms and Special functions, 10(2000), 93-100.

http://science.icu.ac.jp/~morimoto/

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-75.