Wavelet Analysis on the Sphere: Spheroidal Wavelets

Wavelets are a powerful tool in many fields, such as numerical analysis, signal and image processing or data analysis. As the title suggests, the main objective of this book is to present the construction of wavelets for functions defined on the sphere.

The book is organised in 6 chapters. The first one is an introduction to the topic and a guide throughout the next five chapters. In chapters 2,3, and 4, the authors study some mathematical tools useful in wavelet analysis. First of all, they present the notion and basic properties of orthogonal polynomials. They also introduce three different methods for their construction: Rodrigues formula, recurrence rules and orthogonal polynomials as solutions to ODEs. Then they apply these results to the construction of classical families of orthogonal polynomials such as Legendre, Laguerre, Hermite, Chebyshev and Gegenbauer. In the third chapter, they present the homogeneous polynomials and their interaction with harmonic analysis on the sphere. They begin with the study of the spherical harmonics in $\mathbb{S}^{2}$ as the solutions of the Laplace equation and then they develop the general theory. Chapter 4 is devoted to the study of special functions, such as beta, gamma, Bessel or Legendre functions among others. They detail their definitions and properties and provide some graphic illustrations as well.

The last two chapters focus on wavelets. First, the authors study wavelets related to orthogonal polynomials such as Chebyshev, Gegenbauer or Cauchy wavelets; next they concentrate on spherical wavelets. Finally in Chapter 6, they show some examples of wavelets' applications to numerical solutions of PDEs, integrodifferential equations, image and signal processing and time-series processing.

Blanca Fernández
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