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TitleIntroduction to Hyperfunctions and Their Integral Transforms: An Applied and Computational Approach
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Table of Contents
Introduction to Hyperfunctions and Their Integral Transforms: An Applied and Computational Approach
1 Introduction to Hyperfunctions
	1.1 Generalized Functions
	1.2 The Concept of a Hyperfunction
	1.3 Properties of Hyperfunctions
		1.3.1 Linear Substitution
		1.3.2 Hyperfunctions of the Type f(φ(x))
		1.3.3 Differentiation
		1.3.4 The Shift Operator as a Differential Operator
		1.3.5 Parity, Complex Conjugate and Realness
		1.3.6 The Equation φ(x)f(x) = h(x)
	1.4 Finite Part Hyperfunctions
	1.5 Integrals
		1.5.1 Integrals with respect to the Independent Variable
		1.5.2 Integrals with respect to a Parameter
	1.6 More Familiar Hyperfunctions
		1.6.1 Unit-Step, Delta Impulses, Sign, Characteristic Hyperfunctions
		1.6.2 Integral Powers
		1.6.3 Non-integral Powers
		1.6.4 Logarithms
		1.6.5 Upper and Lower Hyperfunctions
		1.6.6 The Normalized Power x^α_+/Γ(α + 1)
		1.6.7 Hyperfunctions Concentrated at One Point
2 Analytic Properties
	2.1 Sequences, Series, Limits
	2.2 Cauchy-type Integrals
	2.3 Projections of Functions
		2.3.1 Functions Satisfying the H¨older Condition
		2.3.2 Projection Theorems
		2.3.3 Convergence Factors
		2.3.4 Homologous and Standard Hyperfunctions
	2.4 Projections of Hyperfunctions
		2.4.1 Holomorphic and Meromorphic Hyperfunctions
		2.4.2 Standard Defining Functions
			Sato’s Decomposition Theorems
			Perfect Hyperfunctions and Their Standard Defining Functions
			Extension to Non-Perfect Hyperfunctions
			Strong Defining Functions
		2.4.3 Micro-analytic Hyperfunctions
		2.4.4 Support, Singular Support and Singular Spectrum
	2.5 Product of Hyperfunctions
		2.5.1 Product of Upper or Lower Hyperfunctions
		2.5.2 Products in the Case of Disjoint Singular Supports
		2.5.3 The Integral of a Product
		2.5.4 Hadamard’s Finite Part of an Integral
	2.6 Periodic Hyperfunctions and Their Fourier Series
	2.7 Convolutions of Hyperfunctions
		2.7.1 Definition and Existence of the Convolution
		2.7.2 Sufficient Conditions for the Existence of Convolutions
		2.7.3 Operational Properties
		2.7.4 Principal Value Convolution
	2.8 Integral Equations I
3 Laplace Transforms
	3.1 Loop Integrals
	3.2 The Two-Sided Laplace Transform
		3.2.1 The Classical Laplace Transform
	3.3 Laplace Transforms of Hyperfunctions
	3.4 Transforms of some Familiar Hyperfunctions
		3.4.1 Dirac Impulses and their Derivatives
		3.4.2 Non-negative Integral Powers
		3.4.3 Negative Integral Powers
		3.4.4 Non-integral Powers
		3.4.5 Powers with Logarithms
		3.4.6 Exponential Integrals
		3.4.7 Transforms of Finite Part Hyperfunctions
	3.5 Operational Properties
		3.5.1 Linearity
		3.5.2 Image Translation Rule
		3.5.3 The Multiplication or Image Differentiation Rule
		3.5.4 Similarity Rule
		3.5.5 Differentiation Rule
		3.5.6 Integration Rule
		3.5.7 Original Translation Rule
		3.5.8 Linear Substitution Rules
	3.6 Inverse Laplace Transforms and Convolutions
		3.6.1 Inverse Laplace Transforms
			The Inversion Formulas
			Laplace hyperfunctions
		3.6.2 The Convolution Rule
		3.6.3 Fractional Integrals and Derivatives
	3.7 Right-sided Laplace Transforms
	3.8 Integral Equations II
		3.8.1 Volterra Integral Equations of Convolution Type
		3.8.2 Convolution Integral Equations over an Infinite Range
4 Fourier Transforms
	4.1 Fourier Transforms of Hyperfunctions
		4.1.1 Basic Definitions
		4.1.2 Connection to Laplace Transformation
	4.2 Fourier Transforms of Some Familiar Hyperfunctions
	4.3 Inverse Fourier Transforms
		4.3.1 Reciprocity
	4.4 Operational Properties
		4.4.1 Linear Substitution Rule
		4.4.2 Shift-Rules
		4.4.3 Complex Conjugation and Realness
		4.4.4 Differentiation and Multiplication Rule
		4.4.5 Convolution Rules
	4.5 Further Examples
	4.6 Poisson’s Summation Formula
	4.7 Application to Integral and Differential Equations
		4.7.1 Integral Equations III
		4.7.2 Heat Equation and Weierstrass Transformation
5 Hilbert Transforms
	5.1 Hilbert Transforms of Hyperfunctions
		5.1.1 Definition and Basic Properties
		5.1.2 Operational Properties
		5.1.3 Using Fourier Transforms
	5.2 Analytic Signals and Conjugate Hyperfunctions
	5.3 Integral Equations IV
6 Mellin Transforms
	6.1 The Classical Mellin Transformation
	6.2 Mellin Transforms of Hyperfunctions
	6.3 Operational Properties
		6.3.1 Linearity
		6.3.2 Scale Changes
		6.3.3 Multiplication by (log x)^n
		6.3.4 Multiplication by x^μ, μ ∈ \mathbb{C}
		6.3.5 Reflection
		6.3.6 Differentiation Rules
		6.3.7 Integration Rules
	6.4 Inverse Mellin Transformation
	6.5 \mathcal{M}-Convolutions
		6.5.1 Reciprocal Integral Transforms
		6.5.2 Transform of a Product and Parseval’s Formula
	6.6 Applications
		6.6.1 Dirichlet’s Problem in a Wedge-shaped Domain
		6.6.2 Euler’s Differential Equation
		6.6.3 Integral Equations V
		6.6.4 Summation of Series
7 Hankel Transforms
	7.1 Hankel Transforms of Ordinary Functions
		7.1.1 Genesis of the Hankel Transform
		7.1.2 Cylinder Functions
		7.1.3 Lommel’s Integral
		7.1.4 MacRobert’s Proof
		7.1.5 Some Hankel Transforms of Ordinary Functions
		7.1.6 Operational Properties
			Parseval’s Relation
			The Similarity and Division Rule
			Differentiation Rules
	7.2 Hankel Transforms of Hyperfunctions
		7.2.1 Basic Definitions
		7.2.2 Transforms of some Familiar Hyperfunctions
		7.2.3 Operational Properties
			The Similarity Rule
			The Division Rule
			The Differentiation Rule
	7.3 Applications
A Complements
	A.1 Physical Interpretation of Hyperfunctions
		A.1.1 Flow Fields and Holomorphic Functions
		A.1.2 Polya fields and Defining Functions
	A.2 Laplace Transforms in the Complex Plane
		A.2.1 Functions of Exponential Type
		A.2.2 Laplace Hyperfunctions and their Transforms
	A.3 Some Basic Theorems of Function Theory
		A.3.1 Interchanging Infinite Series with Improper Integrals
		A.3.2 Reversing the Order of Integration
		A.3.3 Defining Holomorphic Functions by Series and Integrals
B Tables
	Convolution Properties of Hyperfunctions
	Operational Rules for the Laplace Transformation
	Some Laplace Transforms of Hyperfunctions
	Operational Rules for the Fourier Transformation
	Some Fourier Transforms of Hyperfunctions
	Operational Rules for the Hilbert Transformation
	Some Hilbert Transforms of Hyperfunctions
	Operational Rules for the Mellin Transformation
	Some Mellin Transforms of Hyperfunctions
	Operational Rules for the Hankel Transformation
	Some Hankel Transforms of order ν of Hyperfunctions
List of Symbols

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