Abstract
Abstract Many problems arising in geomathematics are of tensorial nature and spherical geometry. An important tool in dealing with such problems is the theory of tensor spherical harmonics. We present in this book a theory of tensor spherical harmonics generalizing in canonical way our approach to vector spherical harmonics. A striking point is that the tensor spherical harmonics are generated from the scalar ones by use of certain operators mapping scalar functions to tensor fields. As a matter of fact, these formulations (being as always independent of any choice of spherical coordinates) offer the possibility of extending all essential results known for scalar spherical harmonics to the tensorial case, including the definition of a tensorial Beltrami operator, the addition theorem, and tensorial versions of the Funk-Hecke formula. Furthermore, our appoach allows a straightforward transition of scalar approximation methods to the tensorial framework.
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