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E-raamat: Generalized Harmonic Analysis and Wavelet Packets: An Elementary Treatment of Theory and Applications [Taylor & Francis e-raamat]

  • Formaat: 320 pages
  • Ilmumisaeg: 07-Mar-2001
  • Kirjastus: Taylor & Francis Ltd
  • ISBN-13: 9780429081910
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  • Taylor & Francis e-raamat
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Generalized Harmonic Analysis and Wavelet Packets: An Elementary Treatment of Theory and ApplicationsSuurem pilt
  • Formaat: 320 pages
  • Ilmumisaeg: 07-Mar-2001
  • Kirjastus: Taylor & Francis Ltd
  • ISBN-13: 9780429081910
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9780429081910
Treats transmutation operators associated with the Bessel operator and explores many of their properties, offering application illustrations and problems. Considers the Riemann-Liouville and the Weyl integral transforms associated with the Bessel operator, and uses these transforms to construct a harmonic analysis associated with the Bessel operator. Chapter topics include generalized convolution products associated with the Bessel operator, the Fourier-Bessel transform, infinitely divisible probabilities and central limit theorem, and wavelet packets associated with the Bessel operator. For graduate students and researchers working in special functions, Fourier analysis, hypergroup and operator theories, and mathematical physics. The author is professor in the Mathematical Department of the Faculty of Sciences of Tunis, Tunisia. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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
Khalifa Trimeche
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