Contents

List of Figures

List of Tables

Preface

1. Introduction

2. Review of orthogonal polynomials

2.1 Introduction

2.2 Generalities

2.3 Orthogonal polynomials via a three-level recurrence

2.4 Darboux–Christoffel rule

2.5 Continued fractions

2.6 Orthogonal polynomials via Rodrigues rule

2.7 Orthogonal polynomials via differential equations

2.8 Some classical orthogonal polynomials

2.8.1 Legendre polynomials

2.8.2 Laguerre polynomials

2.8.3 Hermite polynomials

2.8.4 Chebyshev polynomials

2.8.5 Gegenbauer polynomials

2.9 Conclusion

3. Homogenous polynomials and spherical harmonics

3.1 Introduction

3.2 Spherical Laplace operator

3.3 Some direct computations on S2

3.4 Homogenous polynomials

3.5 Spherical harmonics

3.6 Fourier transform of spherical harmonics

3.7 Zonal functions

3.8 Conclusion

4. Review of special functions

4.1 Introduction

4.2 Classical special functions

4.2.1 Euler’s G function

4.2.2 Euler’s beta function

4.2.3 Theta function

4.2.4 Riemann zeta function

4.2.5 Hypergeometric function

4.2.6 Legendre function

4.2.7 Bessel function

4.2.8 Hankel function

4.2.9 Mathieu function

4.2.10 Airy function

4.3 Hankel–Bessel transform

5. Spheroidal-type wavelets

5.1 Introduction

5.2 Wavelets on the real line

5.3 Chebyshev wavelets

5.4 Gegenbauer wavelets

5.5 Hermite wavelets

5.6 Laguerre wavelets

5.7 Bessel wavelets

5.8 Cauchy wavelets

5.9 Spherical wavelets

6. Some applications

6.1 Introduction

6.2 Wavelets for numerical solutions of PDEs

6.3 Wavelets for integrodifferential equations

6.4 Wavelets in image and signal processing

6.5 Wavelets for time-series processing

Bibliography