Spherical harmonics and approximations on the unit sphere. An introduction
The book concentrates on the theory of spherical harmonics on the unit sphere of a general \(d\)-dimensional Euclidian space. It summarizes the results related to Legendre and Gegenbauer polynomials as well as the theory of differentiation and integration over the \(d\)-dimensional unit sphere and the associated function spaces. Applications of such theoretical constructions to the theory of approximation and numerical quadrature together with description of spectral methods for several well known problems are presented in the remaining part of the book. The style of material presentation was deliberately chosen by the authors with the aim to make the theory described in the book accessible to a wider audience of readers with only some basic knowledge in the functional analysis and measure theory. The book consists of a preface, six chapters, a references list and an index. In Chapter 1, the authors supply the reader with some basic notation and facts. Chapter 2 presents the theory of spherical harmonics from the point of view of the linear function space defined on a unit sphere \(S^d\). Spherical harmonics are introduced as a restriction of harmonic homogeneous polynomials to the sphere. After some basic definitions stated at the beginning of this chapter the authors move on to a description of several important properties of the spherical harmonics such as an addition theorem and the Funk-Hecke formula. And then they define the projection operator onto spherical harmonics function subspaces, which proves to be a useful tool in proving various properties of the spherical harmonics and establishing relations between several polynomial spaces. These topics are discussed later in the chapter. After that the authors move to the properties and representation formulas for Legendre and Gegenbauer polynomials. The rest of the chapter is devoted to the discussion of associated Legendre functions and their role in generating orthogonal bases for spherical harmonic function spaces. Chapter 3 focuses on formulas of differentiation and integration involving spherical harmonics and their properties. Here, the authors derive representation formulas for the Laplace-Beltrami operator, which is defined by the restriction of the Laplace operator to the unit sphere. Using an exact formula for the representation of the mentioned operator trough coordinates and derivatives with respect to the variable \(x\) in \(\mathbb {R}^d\), it is shown that spherical harmonics are eigenfunctions of the Laplace-Beltrami operator. This fact helps to derive several differentiation formulas presented in the chapter, which are related to the spherical harmonics. Further, the authors review some basic properties of harmonic functions for a general dimension \(d\geq 3\) and use them along with the Funk-Hecke formula to justify some integral identities presented here as well. Then, the authors finally introduce Sobolev spaces over the unit sphere via expansions in terms of an orthogonal basis of spherical harmonics. The chapter is concluded with a discussion of the concept of positive definite functions and their connections to the study of meshless discretization methods. Chapter 4 gives the reader a more detailed account of approximation theory in \(S^d\) based on spherical harmonics which was initially described in Chapter 2. In this chapter, the authors restrict their attention to the approximation of real-valued functions by spherical polynomials in \(S^2\) and in the unit disk in \(\mathbb {R}^2\). They summarize the available results on the best approximation of functions by spherical polynomials in the given spaces. Chapter 5 is devoted to numerical quadratures in \(S^2\) and the unit disk in \(\mathbb {R}^2\). Here, the authors use the convenient approach to constructing quadrature formulas by approximation of a corresponding integral \(I(f)\) as a function of \(f\). They begin with methods based on the application of specific single variable quadrature schemes to the representation of \(I(f)\) as a double integral using spherical coordinates. Methods based on the decomposition of the sphere with the help of a mesh and simple quadrature over the mesh elements are also studied in this chapter. After that the authors turn their attention to high-order methods that generalize the concept of Gaussian quadrature for functions of a single variable. They also mention a simple approach to numerical integrations of empiric data and several methods for integral quadrature with function containing a point singularity. The chapter ends with a discussion of the use of numerical integration to approximate the truncated Laplace expansion on \(S^2\) and the truncated orthogonal polynomial expansion on the unit disk in \(\mathbb{R}^2\). In Chapter 6, three application of the theory described are proposed. The first describes how to solve the Dirichlet problem defined in an open simply-connected region \(\Gamma \subset \mathbb{R}^3\) by converting it to an integral equation over \(S^2\) and then applying a Galerkin method based on the spherical polynomial basis. The second application is aimed to solve the Neumann problem in an open simply-connected region of \(\mathbb{R}^2\). This problem is converted into an equivalent elliptic problem over the unit disk and then numerically solved using the Galerkin method based on polynomials over \(\mathbb{R}^2\). The last part of the chapter contains a discussion of the Galerkin method for a specific Beltrami-type equation.
scientific article; zbMATH DE number 5994018