Some remarks on harmonic projection operators on spheres. (English) Zbl 1376.43009
“Bruno Pini” Mathematical Analysis Seminar 2016. Papers from the seminar, University of Bologna, Bologna, Italy, 2016. Bologna: Università di Bologna, Alma Mater Studiorum. 1-17 (2016).
The author gives a survey of recent work concerning the mapping properties of joint harmonic projection operators
\[
\pi_p:L^2(S^{dn-1})\to V^p
\]
mapping the space of square integrable functions noncomplex and quaternionic spheres onto the eigenspaces of the Laplace-Beltrami operator and of a suitably defined subLaplacian. In particular, the similarities and differences are discussed between the real ( \(V^p={\mathcal H^l}\) is the space of spherical harmonics), the complex (\(V^p={\mathcal V^{l,l'}}\) is the space of complex spherical harmonics) and the quaternionic (\(V^p\) is the joint eigenspace of the Laplace Beltrami operator) framework.
For the entire collection see [Zbl 1367.35008].
For the entire collection see [Zbl 1367.35008].
Reviewer: Lubomira Softova (Salerno)
MSC:
43A85 | Harmonic analysis on homogeneous spaces |
35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
33C55 | Spherical harmonics |
43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |