Two spaces of generalized functions based on harmonic polynomials. (English) Zbl 0585.46031
Polynômes orthogonaux et applications, Proc. Laguerre Symp., Bar-le- Duc/France 1984, Lect. Notes Math. 1171, 164-173 (1985).
Summary: [For the entire collection see Zbl 0572.00007.]
Two spaces of generalized functions on the unit sphere \(\Omega^{q- 1}\subset {\mathbb{R}}^ q\) are introduced. Both types of generalized functions can be identified with suitable classes of harmonic functions. They are projective and inductive limits of Hilbert spaces. Several natural classes of continuous and continuously extendible operators are discussed: Multipliers, differentiations, harmonic contractions/expansions and harmonic shifts. The latter two classes of operators are ”parametrized” by the full affine semigroup on \({\mathbb{R}}^ n\).
Two spaces of generalized functions on the unit sphere \(\Omega^{q- 1}\subset {\mathbb{R}}^ q\) are introduced. Both types of generalized functions can be identified with suitable classes of harmonic functions. They are projective and inductive limits of Hilbert spaces. Several natural classes of continuous and continuously extendible operators are discussed: Multipliers, differentiations, harmonic contractions/expansions and harmonic shifts. The latter two classes of operators are ”parametrized” by the full affine semigroup on \({\mathbb{R}}^ n\).
MSC:
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 46F10 | Operations with distributions and generalized functions |
| 31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
| 20G05 | Representation theory for linear algebraic groups |
