Computational harmonic analysis for tensor fields on the two-sphere
The classical spherical harmonics can be defined as an orthogonal system which diagonalizes any rotationally invariant differential operator. The latter can be written as sum of powers of the Laplacian. Due to orthogonality one can define the scalar spherical harmonic transform of square integrable functions on the sphere. The authors develop a similar system of functions for vector fields on the sphere in the three-dimensional Euclidean space. Therefore they generalize the Laplace operator and the angular momentum operator to act on tagential vector fields. Following the construction of the spin-s harmonics [cf. \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro}, Representations of the rotation and Lorentz groups and their applications (1963; Zbl 0108.22005); \textit{J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich} and \textit{E. C. G. Sudarshan}, Spin-s spherical harmonics and eth (1967; Zbl 0155.57402)] they define the monopole harmonics of type \(n\), degree \(l\), and order \(m\) as solutions of a certain eigenvalue problem. It is further proved that this set of functions satisfies an orthogonality relation. This leads to the tensor spherical transform. An explicit definition of the tensor spherical harmonics in terms of linear combinations of products of type \[ {1 \over{(\sin \theta)^{|n|}}} \cdot Y_{p,m} \] is given, where \(Y_{p,m}\) denote the scalar spherical harmonics. The authors then define spaces of band-limited tensors and give an appropriate sampling theorem. The paper closes with the discussion of an algorithm to compute the tensor spherical transform and some numerical examples.
scientific article; zbMATH DE number 1526482
scientific article