We will use the spherical harmonic addition theorem for, (7) (Throughout our paper we use real and Schmidt semi-normalised, with for positive m, and for negative m. These have mean-square value over the sphere of This gives (8) where the spatial functions are the harmonic functions (orthogonal over the sphere) (9) and the moments are given by (10).
In treating chains or clusters of spheres, a common approach is to use the spherical-harmonic addition theorem to relate the multipole expansion coefficients between different spheres.
From the Jacobi-Anger expansion in (10) and the Addition Theorem of spherical harmonics in (A.2), the spherical harmonic expansion of plane waves can be written as in (11) [22] (10).
This theorem of spherical geometry provides an elegant way to define the motion of the lithospheric plates across Earth's surface.
Although Book III contains the first-known extension of Menelaus's theorem for spherical triangles, it is quite possible that the theorem was already known and Menelaus simply transmitted it to later generations.
The translational addition theorem for spherical vector wave functions is employed to impose the zero traction boundary conditions, leading to frequency equations in the form of exact determinantal equations involving spherical Bessel functions and Wigner 3j symbols.
According to the Addition Theorem of spherical harmonics, (A.2).
Applying the Addition Theorem for spherical harmonics (A.2) to (14) transforms the above equation to (15).
Making use of (5) and the Addition Theorem for spherical harmonics in (A.2), the coefficients can be written in a closed-form expression as follows: (8).
This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis, as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.
Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujanʼs Master Theorem for the spherical Fourier transform of a spherical Fourier series.
