Old and New on the Laplace-Beltrami Derivative

The (strong) Laplace-Beltrami derivative and related operators appear quite frequently in many problems involving Fourier series, spherical harmonics, approximation of functions and smoothness. This article intends to provide an overview with updates on such operators. While most of the results concerning the Laplace-Beltrami derivative found in the literature are stated in the three-dimensional setting, our approach includes higher-dimensional spheres. We present proofs for results which are labeled as known but are just mentioned elsewhere. Spherical moduli of smoothness and decay rates for eigenvalues of integral operators are also among the covered topics.