Wavelet analysis on the sphere. Spheroidal wavelets (Q2836176)

This textbook consists of two parts. The first part (Chapters 2--4) presents known results on orthogonal polynomials (such as Legendre, Chebyshev, and Gegenbauer polynomials), spherical harmonics, and special functions (such as Euler's gamma function, theta function, and hypergeometric functions). These functions are illustrated in many figures.NEWLINENEWLINEIn the second part (Chapters 5--6) which contains many errors, the authors try to define wavelets based on orthogonal polynomials. These piecewise continuous wavelets are only transformed orthogonal polynomials restricted to certain small intervals. Hence the corresponding wavelet expansions are only modified expansions into orthogonal polynomials. The wavelet coefficients can be computed only by inner products, but not by a recursive decomposition algorithm. For a good description of spherical wavelets and related applications in geosciences, the reviewer recommends to use [\textit{V. Michel}, Lectures on constructive approximation. Fourier, spline, and wavelet methods on the real line, the sphere, and the ball. New York, NY: Birkhäuser (2013; Zbl 1295.65003)]. For a general theory of polynomial wavelets see [\textit{B. Fischer} and \textit{J. Prestin}, Math. Comput. 66, No. 220, 1593--1618 (1997; Zbl 0896.42020)].