The book presents a more comprehensive treatment of transmutation operators associated with the Bessel operator & explores many of their properties. They are fundamental in the complete study of the Bessel harmonic analysis & the Bessel wavelet packets. Many applications of these theories & their generalizations have been injected throughout the text by way of a rich collection of problems & references.

Khalifa Trimeche is presently a Professor at the Mathematical Department of the Faculty of Sciences of Tunis, Tunisia

Introduction

xi

The Normalized Bessel Function of First Kind

1

(24)

Introduction

1

(1)

The bessel function of first kind

2

(6)

Definition

2

(1)

Derivatives and differential equation of the Bessel function Jν

3

(1)

Asymptotic formulas for the Bessel function Jν

4

(1)

The Poisson integral representations of the Bessel function Jν

5

(1)

The Sonine's first integral for the Bessel functions Jα and Jβ

6

(1)

Addition formulas for the Bessel function Jν

7

(1)

Product formulas for the Bessel function Jν

7

(1)

The normalized Bessel function of first kind

8

(7)

Definition

8

(1)

Properties of the function jα(λx)

9

(1)

Integral representations of the function jα (λx)

10

(1)

The Poisson integral representations

10

(2)

The Sonine's first integral

12

(1)

Product formula for the function jα

12

(2)

Useful formulas involving the function jα

14

(1)

Problems

15

(10)

Riemann-Liouville and Weyl Integral Transforms

25

(44)

Introduction

25

(1)

The Riemann-Liouville integral transform

26

(15)

Definition and properties

26

(5)

Inversion of the operator Rα

31

(7)

The Riemann-Liouville integral transform on the spaces Lp([0, + ∞[, dx), 1≤p≤+∞

38

(3)

The Weyl integral transform

41

(10)

Definition and properties

41

(3)

Inversion of the operator Wα

44

(7)

The Weyl integral transform on the space &*(R)

51

(4)

The Weyl transform on the space E'*(R)

55

(1)

The Sonine integral transform and its dual

56

(6)

The Sonine integral transform

56

(3)

The Sonine integral transform on the spaces Lp([0, + ∞[, dx), 1≤p≤+∞

59

(1)

The dual Sonine integral transform

60

(1)

The Sonine transform on the space E'*(R)

61

(1)

Problems

62

(7)

Convolution Product and Fourier-Cosine Transform of Functions, Measures and Distributions

69

(18)

Introduction

69

(1)

Convolution product of functions and distributions

69

(6)

The translation operator

70

(1)

Convolution product of functions

71

(2)

Convolution product of measures

73

(2)

Convolution product of distributions

75

(1)

The Fourier-cosine transform

75

(6)

The Fourier-cosine transform on L1([0, + ∞[, dx)

75

(2)

The Fourier-cosine transform on &*(R) and D*(R)

77

(1)

The Fourier-cosine transform on L2([0, + ∞[, dx)

78

(1)

The Fourier-cosine transform on Mb([0, + ∞[)

79

(1)

The Fourier-cosine transform on E'*(R) and &*(R)

80

(1)

Problems

81

(6)

Generalized Convolution Product Associated with The Bessel Operator

87

(28)

Introduction

87

(1)

Convolution product of radial functions

88

(4)

Definition and properties

88

(1)

Convolution product

89

(3)

Generalized translation operators associated with the Bessel operator

92

(5)

Generalized convolution product associated with the Bessel operator

97

(11)

Generalized convolution product of functions

97

(9)

Generalized convolution product of measures of Mb([0,+∞[)

106

(1)

Generalized convolution product of distributions

107

(1)

Problems

108

(7)

Fourier-Bessel Transform

115

(58)

Introduction

115

(1)

Fourier transform of radial functions

116

(3)

The Fourier-Bessel transform on L1([0, + ∞[, d&mu:α)

119

(8)

The Fourier-Bessel transform on &*(R) and D*(R)

127

(5)

The Fourier-Bessel transform on L2([0, + ∞[, dμα)

132

(7)

The Fourier-Bessel transform on Lp([0, + ∞[, dμα), 1≤p≤2

139

(2)

The Fourier-Bessel transform on Mb([0, + ∞[)

141

(3)

The Fourier-Bessel transform on E'*(R) and &*(R)

144

(6)

Problems

150

(23)

Infinitely Divisible Probabilities And Central Limit Theorem Associated with The Bessel Operator

173

(24)

Introduction

173

(1)

Dispersion of a Probability measure on [0, + ∞[

174

(8)

Generalized quadratic form

174

(2)

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

176

(3)

Levy's theorem

179

(3)

Levy-Khintchine's formula

182

(4)

Convolution semigroups and infinitely divisible probabilities associated with the Bessel operator

186

(5)

Convolution semigroups

186

(1)

Infinitely divisible probabilities associated with the Bessel operator

187

(4)

Central limit theorem associated with the Bessel operator

191

(6)

Continuous Wavelet Transform Associated with The Bessel Operator

197

(42)

Introduction

197

(1)

Classical continuous wavelet transform on [0, + ∞[

198

(5)

Classical wavelets on [0, + ∞[

198

(3)

Classical continuous wavelet transform on [0, + ∞[

201

(2)

Continuous wavelet transform associated with the Bessel operator

203

(15)

Wavelets associated with the Bessel operator

203

(4)

Continuous wavelet transform associated, with the Bessel operator

207

(11)

Inversion formulas for the operators Rα and Wα

218

(5)

Inversion formulas for the operators Rα and Wα using wavelets associated with the Bessel operator

223

(8)

Problems

231

(8)

Wavelet Packets Associated with The Bessel Operator

239

(34)

Introduction

239

(1)

The P-wavelet packet transform associated with the Bessel operator

239

(8)

Plancherel and reconstruction formulas

240

(5)

Calderon's reproducing formula

245

(2)

Scale discrete scaling function associated with the Bessel operator

247

(7)

Modified packet associated with the Bessel operator

254

(6)

S-wavelet packet associated with the Bessel operator

260

(5)

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

265

(3)

Problems

268

(5)

Continuous Linear Wavelet Transform Associated with The Bessel Operator And Its Discretization

273

(20)

Introduction

273

(1)

Linear wavelets associated with the Bessel operator

273

(7)

Linear wavelet packets associated with the Bessel operator

280

(5)

Scale discrete L-scaling function associated with the Bassel operator

285

(8)

Bibliography

293

(10)

Index

303

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.