This book is an enlarged second edition of a monograph published in the Springer AGEM2-Series, 2009. It presents, in a consistent and unified overview, a setup of the theory of spherical functions of mathematical (geo-)sciences. The content shows a twofold transition: First, the natural transition from scalar to vectorial and tensorial theory of spherical harmonics is given in a coordinate-free context, based on variants of the addition theorem, Funk-Hecke formulas, and Helmholtz as well as Hardy-Hodge decompositions. Second, the canonical transition from spherical harmonics via zonal (kernel) functions to the Dirac kernel is given in close orientation to an uncertainty principle classifying the space/frequency (momentum) behavior of the functions for purposes of data analysis and (geo-)application. The whole palette of spherical functions is collected in a well-structured form for modeling and simulating the phenomena and processes occurring in the Earth's system. The result is a work which, while reflecting the present state of knowledge in a time-related manner, claims to be of largely timeless significance in (geo-)mathematical research and teaching.
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From the reviews:
"This book concentrates the introduction of mathematical representation of spherical vector and tensor fields. Vector/tensor spherical harmonics are used throughout mathematics. This book is a valuable reference for scientists and practitioners when facing spherical problems." (Chengshu Wang, Zentralblatt MATH, Vol. 1167, 2009)
"This book concentrates the introduction of mathematical representation of spherical vector and tensor fields. Vector/tensor spherical harmonics are used throughout mathematics. This book is a valuable reference for scientists and practitioners when facing spherical problems." (Chengshu Wang, Zentralblatt MATH, Vol. 1167, 2009)