Generalized Harmonic Analysis and Wavelet Packets An Elementary Treatment of Theory and Applications 1st edition by Khalifa Trimeche ISBN 1482283174 9789056993290

Generalized Harmonic Analysis and Wavelet Packets An Elementary Treatment of Theory and Applications 1st edition by Khalifa Trimeche ISBN 1482283174 9789056993290

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ISBN 10: 1482283174
ISBN 13: 9789056993290
Author: Khalifa Trimeche 

The book presents a more comprehensive treatment of transmutation operators associated with the Bessel operator, and explores many of their properties. They are fundamental in the complete study of the Bessel harmonic analysis and the Bessel wavelet packets. Many applications of these theories and their generalizations have been injected throughout the text by way of a rich collection of problems and references. The results and methods in this book should be of interest to graduate and researchers working in special functions such as Fourier analysis, hypergroup and operator theories, differential equations, probability theory and mathematical physics. Background materials are given in adequate detail to enable a graduate student to proceed rapidly from the very basics of the frontier of research in the area of generalized harmonic analysis and wavelets.

Generalized Harmonic Analysis and Wavelet Packets: An Elementary Treatment of Theory and Applications 1st Table of contents:

1 The Normalized Bessel Function Of First Kind

Introduction

1.I. The Bessel function of first kind

1.1.1. Definition

1.1.2. Derivatives and differential equation of the Bessel function J[sub(v)]

1.1.3. Asymptotic formulas for the Bessel function J[sub(v)]

1.1.4. The Poisson integral representations of the Bessel function J[sub(v)]

1.1.5. The Sonine's first integral for the Bessel functions J[sub(α)] and J[sub(β)]

1.1.6. Addition formulas for the Bessel function J[sub(v)]

1.1.7. Product formulas for the Bessel function J[sub(v)]

1.II. The normalized Bessel function of first kind

1.II.1. Definition

1.II.2. Properties of the function j[sub(α)] (λx)

1.II.3. Integral representations of the function j[sub(α)] (λx)

1.II.3.a. The Poisson integral representations

1.II.3.b. The Sonine's first integral

1.II.4. Product formula for the function j[sub(α)]

1.II.5. Useful formulas involving the function j[sub(α)]

1.III. Problems

2 Riemann-Liouville And Weyl Integral Transforms

Introduction

2.I. The Riemann-Liouville integral transform

2.1.1. Definition and properties

2.1.2. Inversion of the operator R[sub(α)]

2.1.3. The Riemann-Liouville integral transform on the spaces L[sup(p)]([0, + ∞[, dx), 1≤p≥+

2.II. The Weyl integral transform

2.II.1. Definition and properties

2.II.2. Inversion of the operator W[sup(α)]

2.III. The Weyl integral transform on the space &[sub(*)] (R )

2.IV. The Weyl transform on the space &'[sub(*)] (R)

2.V. The Sonine integral transform and its dual

2.V.I. The Sonine integral transform

2.V.2. The Sonine integral transform on the spaces L[sup(p)]([0, + ∞[, dx), 1≤p≥+∞

2.V.3. The dual Sonine integral transform

2.V.4. The Sonine transform on the space &'[sub(*)] (R)

2.VI. Problems

3 Convolution Product And Fourier-Cosine Transform Of Functions, Measures And Distributions

Introduction

3.I. Convolution product of functions and distributions

3.1.1. The translation operator

3.1.2. Convolution product of functions

3.1.3. Convolution product of measures

3.1.4. Convolution product of distributions

3.II. The Fourier-cosine transform

3.II.1. The Fourier-cosine transform on L[sup(1)]([0, +∞[, dx)

3.II.2. The Fourier-cosine transform on &[sub(*)](R) and D[sub(*)](R)

3.II.3. The Fourier-cosine transform on L[sup(2)]([0, +∞[, dx)

3.II.4. The Fourier-cosine transform on M[sup(b)]([0, + ∞[)

3.II.5. The Fourier-cosine transform on &'[sub(*)](R) and &'[sub(*)](R)

3.III. Problems

4 Generalized Convolution Product Associated With The Bessel Operator

Introduction

4.I. Convolution product of radial functions

4.1.1. Definition and properties

4.1.2. Convolution product

4.II. Generalized translation operators associated with the Bessel operator

4.III. Generalized convolution product associated with the Bessel operator

4.III.1. Generalized convolution product of functions

4.III.2. Generalized convolution product of measures of M[sup(b)]([0,+∞[)

4.III.3 Generalized convolution product of distributions

4.IV. Problems

5 Fourier-Bessel Transform

Introduction

5.I. Fourier transform of radial functions

5.II. The Fourier-Bessel transform on L[sup(1)]([0, + ∞[, dμ[sub(α)])

5.III. The Fourier-Bessel transform on &[sub(*)](R) and D[sub(*)] (R)

5.IV. The Fourier-Bessel transform on L[sup(2)]([0, + ∞[, dμ[sub(α)])

5.V. The Fourier-Bessel transform on L[sup(p)]([0, + ∞[, dμ[sub(α)]), 1≥p≤2

5.VI. The Fourier-Bessel transform on M[sup(b)]([0, + ∞[)

5.VII. The Fourier-Bessel transform on &'[sub(*)](R) and &'[sub(*)](R)

5.VIII. Problems

6 Infinitely Divisible Probabilities And Central Limit Theorem Associated With The Bessel Operator

Introduction

6.I. Dispersion of a probability measure on [0, + ∞[

6.1.1. Generalized quadratic form

6.1.2. Dispersion of a probability measure on [0, + ∞[

6.1.3. Levy's theorem

6.II. Levy-Khintchine's formula

6.III. Convolution semigroups and infinitely divisible probabilities associated with the Bessel oper

6.III.1. Convolution semigroups

6.III.2. Infinitely divisible probabilities associated with the Bessel operator

6.IV. Central limit theorem associated with the Bessel operator

7 Continuous Wavelet Transform Associated With The Bessel Operator

Introduction

7.I. Classical continuous wavelet transform on [0, + ∞[

7.1.1. Classical wavelets on [0, + ∞[

7.1.2. Classical continuous wavelet transform on [0, + ∞[

7.II. Continuous wavelet transform associated with the Bessel operator

7.II.1. Wavelets associated with the Bessel operator

7.II.2. Continuous wavelet transform associated, with the Bessel operator

7.III. Inversion formulas for the operators R[sub(α)] and W[sub(α)]

7.IV. Inversion formulas for the operators R[sub(α)] and W[sub(α)] using wavelets associated with

7.V. Problems

8 Wavelet Packets Associated With The Bessel Operator

Introduction

8.I. The P-wavelet packet transform associated with the Bessel operator

8.1.1. Plancherel and reconstruction formulas

8.1.2. Calderon's reproducing formula

8.II. Scale discrete scaling function associated with the Bessel operator

8.III. Modified packet associated with the Bessel operator

8.IV. S-wavelet packet associated with the Bessel operator

8.V. Multiresolution analysis by means of wavelet packets associated with the Bessel operator

8.VI. Problems

9 Continuous Linear Wavelet Transform Associated With The Bessel Operator And Its Discretization

Introduction

9.I. Linear wavelets associated with the Bessel operator

9.II. Linear wavelet packets associated with the Bessel operator

9.III. Scale discrete L-scaling function associated with the Bessel operator

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Tags: Khalifa Trimeche, Generalized Harmonic, Wavelet Packets, Elementary Treatment