Spherical functions on Euclidean space

Spherical Functions on Euclidean Space

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space En = G/K where G is the semidirect product Rn · K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)–spherical functions by a certain affine algebraic variety, and of the positive definite on...

Spherical Submanifolds of a Euclidean Space

A1 and A2 being arbitrary constants. A natural generalization of this question to higher dimensions could be: ‘given an isometric immersion ψ : Mn → Rn+2 of a compact n-dimensional Riemannian manifold (Mn, g), obtain conditions for ψ(Mn) ⊂ Sn+1(c), where Sn+1(c) is the sphere of constant curvature c’. We write ψT , ψ⊥ as tangential and normal components of the position vector ψ in Rn+p and show...

On images of continuous functions from a compact manifold to Euclidean space

Spherical Functions and Spherical Laplace Transform on Ordered Symmetric Space

Let G=H be a semisimple globally hyperbolic symmetric space and let ' be a H-spherical function on G=H. We derive an expansion formula for ' similar to the Harish-Chandra formula for spherical functions on a Riemannian symmetric space. We use this result to analytically continuate the spherical functions in the parameters. A functional equation for ' is derived and then used to invert the spher...

Harmonic Morphisms from Three-dimensional Euclidean and Spherical Space Forms

This paper gives a description of all harmonic morphisms from a threedimensional non-simply-connected Euclidean and spherical space form to a surface, by extending the work of Baird-Wood [4, 5] who dealt with the simply-connected case; namely we show that any such harmonic morphism is the composition of a “standard” harmonic morphism and a weakly conformal map. To complete the description we li...