Fractional Integral Transforms: Theory and Applications

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
CHAPTER 1: Introduction and Preliminaries
	1.1. NOTATION
	1.2. SPECIAL FUNCTIONS AND ORTHOGONAL POLYNOMIALS
		1.2.1. The Gamma Function
		1.2.2. The Beta Function
		1.2.3. The Hermite Polynomials Hn(x)
		1.2.4. The Laguerre Polynomials Lɑn(x)(ɑ > – 1)
		1.2.5. The Jacobi Polynomials Pn(ɑ,β)(x)(ɑ, β > – 1)
		1.2.6. The Bessel Functions
		1.2.7. The Mittag–Leffler function
		1.2.8. The Hypergeometric and q-Hypergeometric Functions
	1.3. NON-ORTHOGONAL BASES AND FRAMES IN A HILBERT SPACE
		1.3.1. Non-orthogonal Bases and Frames
		1.3.2. Reproducing-Kernel Hilbert Spaces
	1.4. SHIFT-INVARIANT SPACES
	1.5. GENERALIZED FUNCTIONS AND DISTRIBUTIONS
		1.5.1. Testing-Function Spaces and Their Duals
		1.5.2. Spaces of Generalized Functions
		1.5.3. A Special Type of Generalized Functions
	1.6. SAMPLING AND THE PALEY-WIENER SPACE
	1.7. POISSON SUMMATION FORMULA
	1.8. UNCRTAINTY PRINCIPLE
CHAPTER 2: Integral Transformations
	2.1. INTRODUCTION AND BRIEF HISTORY
	2.2. WHAT IS AN INTEGRAL TRANSFORM?
	2.3. EXAMPLES OF INTEGRAL TRANSFORMS
		2.3.1. One-Dimensional Integral Transforms
		2.3.2. Higher Dimensional Transforms
		2.3.3. Special Cases of Higher Dimensional Transforms
	2.4. GENERAL PROPERTIES OF INTEGRAL TRANSFORMATIONS
	2.5. WHY INTEGRAL TRANSFORMS?
CHAPTER 3: Fractional Integral Transforms
	3.1. INTRODUCTION
	3.2. PRELUDE TO FRACTIONAL INTEGRAL TRANSFORMS
		3.2.1. The Fractional Fourier Transform
		3.2.2. The Fractional Hankel Transform
	3.3. GENERAL CONSTRUCTION OF FRACTIONAL INTEGRAL TRANSFORMS
		3.3.1. Examples of the General Construction
		3.3.2. Fractional Integral Transforms Associated With the Jacobi Polynomials
	3.4. FRACTIONAL DERIVATIVES AND INTEGRALS VERSUS FRACTIONAL INTEGRAL TRANSFORMS
	3.5. OTHER FRACTIONAL INTEGRAL TRANSFORMS
CHAPTER 4: The Fractional Fourier Transform (FrFT)
	4.1. HISTORICAL OVERVIEW
	4.2. PRELIMINARIES
	4.3. OPERATIONAL CALCULUS
		4.3.1. Convolution Theorem
		4.3.2. Poisson Summation Formula for the Fractional Fourier Transform
		4.3.3. Sampling Theorem for the Fractional Fourier Transform
		4.3.4. The Wigner Distribution
	4.4. THE FRACTIONAL HILBERT TRANSFORM
	4.5. FRACTIONAL TIME-FREQUENCY REPRESENTATIONS
		4.5.1. Fractional Wigner Distributions
		4.5.2. Fractional Time and Frequency Shifts
		4.5.3. The Fractional Cross-Ambiguity Function
		4.5.4. Fractional Windowed (Sliding-Window)-Fourier Transform
	4.6. UNCERTAINTY PRINCIPLE FOR THE FRACTIONAL FOURIER TRANSFORM
	4.7. FRACTIONAL FOURIER TRANSFORM OF GENERALIZED FUNCTIONS
		4.7.1. The Embedding Method
		4.7.2. The Space of Boehmians
		4.7.3. The Algebraic Method
	4.8. APPLICATIONS OF THE FRACTIONAL FOURIER TRANSFORM
CHAPTER 5: Shift-Invariant and Sampling Spaces of the Fractional Fourier Transform
	5.1. INTRODUCTION
	5.2. BASIC DEFINITIONS
	5.3. DISCRETE FRACTIONAL FOURIER TRANSFORM AND CONVOLUTION
		5.3.1. Discrete Fractional Fourier Transform
		5.3.2. Fractional Convolution
	5.4. SHIFT-INVARIANCE IN THE FRFT DOMAIN
	5.5. THE FRACTIONAL ZAK TRANSFORM
	5.6. APPLICATIONS: FRACTIONAL DELAY FILTERING
CHAPTER 6: Two-Dimensional Coupled Fractional Fourier Transform (CFrFT)
	6.1. INTRODUCTION
	6.2. FRACTIONAL FOURIER TRANSFORM IN HIGHER DIMENSIONS
		6.2.1. The Direct Product Representation
		6.2.2. Metaplectic Representation
	6.3. THE TWO-DIMENSIONAL FRACTIONAL FOURIER TRANSFORM
		6.3.1. Complex Hermite Polynomials
		6.3.2. Integral Representation of the Two-Dimensional Fractional Fourier Transform
		6.3.3. Inversion Formula
		6.3.4. Examples
	6.4. ADDITIVE PROPERTY
	6.5. CONVOLUTION THEOREM
	6.6. POISSON SUMMATION FORMULA
	6.7. A SPACE OF BANDLIMITED SIGNALS AND ITS SAMPLING THEOREM
		6.7.1. Space of Bandlimited Signals
		6.7.2. Sampling Theorems
		6.7.3. Examples and Sampling Points Configuration
	6.8. THE COUPLED FRACTIONAL FOURIER TRANSFORM OF GENERALIZED FUNCTIONS
	6.9. THE GYRATOR TRANSFORM
		6.9.1. Motivation and Definitions
		6.9.2. Elementary Properties of the Gyrator Transform
CHAPTER 7: The Two-Dimensional Fractional Fourier Transform and The Wigner Distribution
	7.1. INTRODUCTION
	7.2. THE WIGNER DISTRIBUTION
	7.3. FOUR-DIMENSIONAL ROTATIONS
	7.4. THE FOUR-DIMENSIONAL WIGNER DISTRIBUTION
	7.5. THE FOUR-DIMENSIONAL WIGNER DISTRIBUTION AND THE COUPLED FRACTIONAL FOURIER TRANSFORM
CHAPTER 8: Short-Time Coupled Fractional Fourier Transform and Uncertainty Relations
	8.1. INTRODUCTION AND NOTATION
	8.2. PROPERTIES OF THE COUPLED FRACTIONAL FOURIER TRANSFORM
	8.3. CONVOLUTION AND EXTENSION OF THE COUPLED FRACTIONAL FOURIER TRANSFORM
	8.4. SHORT-TIME COUPLED FRACTIONAL FOURIER TRANSFORM
	8.5. PROPERTIES OF THE SHORT-TIME COUPLED FRACTIONAL FOURIER TRANSFORM
	8.6. UNCERTAINTY PRINCIPLE
CHAPTER 9: The Linear Canonical Transform (LCT)
	9.1. INTRODUCTION AND HISTORICAL OVERVIEW
	9.2. DEFINITIONS AND SPECIAL CASES OF THE LINEAR CANONICAL TRANSFORM
	9.3. PROPERTIES OF THE LINEAR CANONICAL TRANSFORM
		9.3.1. Basic Properties
		9.3.2. Convolution Theorems
		9.3.3. Additive Property of the Linear Canonical Transform
		9.3.4. Sampling Theorem
		9.3.5. Eigenfunctions and Eigenvalues
	9.4. THE METAPLECTIC REPRESENTATION AND CONVOLUTION
	9.5. ELEMENTARY PROPERTIES OF THE METAPLECTIC TRANSFORMATIONS
	9.6. TWO-DIMENSIONAL SAMPLING THEOREM FOR THE LINEAR CANONICAL TRANSFORM
		9.6.1. Two-Dimensional LCT in Polar Coordinates
		9.6.2. Sampling Theorem for LCT
CHAPTER 10: The Special Affine Fourier Transform (SAFT)
	10.1. INTRODUCTION AND HISTORICAL REMARKS
	10.2. DEFINITIONS
	10.3. THE OFFSET LINEAR CANONICAL TRANSFORM
	10.4. ELEMENTARY PROPERTIES OF THE SPECIAL AFFINE FOURIER TRANSFORM
	10.5. POISSON SUMMATION FORMULA FOR SAFT
	10.6. CONVOLUTION AND PRODUCT THEOREMS FOR SPECIAL AFFINE FOURIER TRANSFORM
		10.6.1. Modulation and Convolution Operations
		10.6.2. Convolution Theorem
		10.6.3. Product Theorem
	10.7. SHIFT-INVARIANT SPACES FOR THE SPECIAL AFFINE FOURIER TRANSFORM
		10.7.1. Preliminaries
		10.7.2. Discrete Special Affine Fourier Transform
		10.7.3. Riesz Basis for Shift-Invariant Spaces in the SAFT Domain
	10.8. ZAK TRANSFORM ASSOCIATED WITH THE SAFT
	10.9. SHANNON’S SAMPLING THEOREM AND THE SAFT: REINTERPRETATION, EXTENSION AND APPLICATIONS
Appendix
Bibliography
Index