Transmutation Operators and Applications (Trends in Mathematics)

Preface
Contents
Part I Transmutations, Integral Equations and Special Functions
	Some Recent Developments in the Transmutation Operator Approach
		References
	Transmutation Operators and Their Applications
		1 Introduction
		2 Existence and Construction of Transmutations
			2.1 Classical Transmutations
			2.2 Transmutations by Paley–Wiener Theorem
			2.3 Rigged Hilbert Spaces
			2.4 Transmutation with Distinct Spectra
			2.5 Transmutation with Disjoint Spectra
		3 Transmutation for Strings
			3.1 Transmutation for Strings
			3.2 Adding a Potential
			3.3 Examples
		4 Applications
			4.1 The Gelfand-Levitan Theory
			4.2 Gelfand-Levitan Revisited
			4.3 Transmutation Between Orthogonal Polynomials
				4.3.1 Example
			4.4 Direct Reconstruction of the Spectral Function
			4.5 The Lieb and Thirring Constant
			4.6 Gelfand-Levitan for the String
				4.6.1 The Transformation Operator
			4.7 Sampling and Transmutation
			4.8 Computational Spectral Theory
		References
	Hankel Generalized Convolutions with the Associated Legendre Functions in the Kernel and Their Applications
		1 Introduction
		2 Properties and Estimates for the Convolution\'s Kernel
		3 Mapping Properties of the Generalized Convolutions
		4 Integral Transforms Related to the Hankel Polyconvolution
		5 Examples
		References
	Second Type Neumann Series Related to Nicholson\'s and to Dixon–Ferrar Formula
		1 Introduction to Nicholson\'s Formula
		2 Preparation: Euler–Maclaurin Summation Formula, Dirichlet Series and Cahen\'s Formula
		3 Main Results: Accessum per Definitionem
		4 Main Results: The Dixon–Ferrar Formula
		5 Discussion: Open Problems
		References
	On Some Generalizations of the Properties of the Multidimensional Generalized Erdélyi–Kober Operators and Their Applications
		1 Introduction
		2 Generalization of the Properties of the Generalized Erdelyi–Kober Operator
		3 Applications
		Appendix: Integral Transform Composition Method (ITCM) in Transmutation Theory: How It Works
			What is ITCM and How It Works?
			Application of ITCM to Index Shift B–Hyperbolic Transmutations
			Application of Transmutations Obtained by ITCM to Integral Representations of Solutions to Hyperbolic Equations with Bessel Operators
		References
	Alternative Approach to Miller-Paris Transformations and Their Extensions
		1 Introduction and Preliminaries
		2 Miller-Paris Transformations: General Case
		3 Miller-Paris Transformations: Degenerate Case
		References
	Transmutation Operators For Ordinary Dunkl–Darboux Operators
		1 Introduction
		2 Dunkl–Darboux Operators
		3 Darboux Transmutations for High Order Differential Operators
		4 Integral Dunkl–Darboux Transmutations
		5 Transmutation Operators for Dunkl–Darboux Operators in Cherednik Algebra
		6 Recurrence Equations
		7 Transmutation Operators for Dunkl–Darboux Operators in Cherednik Pseudoalgebra
		References
	Theorems on Restriction of Fourier–Bessel and Multidimensional Bessel Transforms to Spherical Surfaces
		1 Introduction
		2 Mixed Fourier–Bessel Transform
		3 N-Dimensional Bessel Transform
		References
	Necessary Condition for the Existence of an Intertwining Operator and Classification of Transmutations on Its Basis
		1 Introduction
		2 Problem Definition
		3 Formulation and Specification of Reverse Statement
		4 Some Convolutions as Transmutation Operators and Their Modifications
		5 Euler Transformation for Hypergeometric Functions as a Transmutation Operator
		References
	Polynomial Quantization on Line Bundles
		1 The Group SL(2,R) and Its Representations
		2 Tensor Products
		3 Hyperboloid of One Sheet
		4 Poisson Transform
		5 Polynomial Quantization
		6 Berezin Transform for Induced Representation
		References
	Fourier–Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations
		1 Introduction
		2 Notation and Definitions
		3 Estimates for the One-Dimensional Case
		4 Estimates for One-Variable Compactly Supported Functions
			4.1 The Case Where the Weight Power Does Not Exceed the Parameter at the Singularity
			4.2 The case where the weight power exceeds the parameter at the singularity
		5 Multi-Dimensional Estimates: The Prototype Case
		6 Estimates for the Case of Several Special Variables
			6.1 Preliminaries
			6.2 Estimates for the General Case
			6.3 The Case of a Single Nonspecial Variable
			6.4 The Case of Absence of Nonspecial Variables
		7 Applications to Singular Equations
			7.1 Estimates of Solutions of Singular Ordinary Differential Equations
			7.2 Estimates of Solutions of Singular Partial Differential Equations
		References
	Inversion of Hyperbolic B-Potentials
		1 Introduction
			1.1 Transmutation Operators
			1.2 A Brief History of the Potentials Operators
			1.3 Basic Definitions
		2 Hyperbolic B-Potentials and Their Properties
			2.1 Definitions of the Hyperbolic B-Potentials
			2.2 Absolute Convergence and Boundedness
		3 Green\'s Second Identity for the Hyperbolic B-Potentials
			3.1 Divergence Theorem for Weighted Nabla Operator
			3.2 Green\'s Second Identities for the γ and for the Hyperbolic B-Potentials
		4 Inversion of the Hyperbolic B-Potentials
			4.1 Method of Approximative Inverse Operators
			4.2 General Poisson Kernel
			4.3 Representation of the Kernel gα,δ
			4.4 Belonging of the (IPi 0,γα)-1,δ to the Class LPγ
			4.5 Theorems About the Inversion of the Hyperbolic B-Potential
		References
	One-Dimensional and Multi-Dimensional Integral Transforms of Buschman–Erdélyi Type with Legendre Functions in Kernels
		1 Buschman–Erdélyi Operators
		2 Multi-Dimensional Integral Transforms of Buschman–Erdélyi Type with Legendre Functions in Kernels
			2.1 Lν,2–Theory and the Inversion Formulas for the Modified H-Transform
			2.2 Representations in the Form of Modified H-Transform
		References
	Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems
		1 Introduction
		2 Domains and Operators
			2.1 Paired Equations
			2.2 Singularities and Distributions
			2.3 Complex Variables and Wave Factorization
		3 Transmutations, Distributions and the Fourier Transform
			3.1 Examples
				3.1.1 Plane Sector
				3.1.2 Standard Cone
				3.1.3 Three-Wedged Pyramid
		4 Potentials Generated by Transmutations
			4.1 General Situation
		5 Boundary Value Problems
		6 Thin Cones
		7 Conclusion
		References
Part II Transmutations in ODEs, Direct and Inverse Problems
	On a Transformation Operator Approach in the Inverse Spectral Theory of Integral and Integro-Differential Operators
		1 Introduction
		2 One-Dimensional Perturbation of a Convolution Operator
			2.1 Historical Notes
			2.2 Statement of the Inverse Problem
			2.3 Transformation Operator
			2.4 Main Nonlinear Integral Equation
			2.5 Solution of a Nonlinear Equation Without Singularity
			2.6 Proof of Theorem 1
			2.7 Solution of Inverse Problem 1
		3 Convolution Integro-Differential Operator
			3.1 Statement of the Inverse Problem and Main Results
			3.2 Transformation Operator
			3.3 The Main Equation
		4 Convolutional Perturbation of the Sturm–Liouville Operator
			4.1 Historical Notes and the Main Result
			4.2 Transformation Operator
			4.3 The Main Equation
		5 Integro-Differential Dirac Systems
			5.1 Statement of the Inverse Problem and Main Results
			5.2 Transformation Operator
			5.3 Characteristic Functions
			5.4 The Main Equation
		References
	Expansion in Terms of Appropriate Functions and Transmutations
		1 Introduction
		2 Presentation of the Class of the Operators and Expansion
		3 Integral Representations
		4 Transmutation
		5 Some Applications
		References
	Transmutation Operators as a Solvability Concept of Abstract Singular Equations
		1 Introduction
		2 Euler–Poisson–Darboux Equation: Bessel Operator Function
		3 Euler–Poisson–Darboux Equation: Bessel Operator Function with Negative Index
		4 The Bessel-Struve Equation: Operator Function Struve
		5 The Legendre Equation: Legendre Operator Function
		6 The Loaded Legendre Equation
		7 Nonlocal Problems
		8 Dirichlet Problem for the Bessel-Struve Equation
		References
	On the Bessel-Wright Operator and Transmutation with Applications
		1 Introduction
		2 The Bessel-Wright Transmutation Operator
		3 Applications
			3.1 The Bessel-Wright Transform
			3.2 The Bessel-Wright Transform Inversion Formula
			3.3 The Bessel-Wright Translation Operator and Its Dual
				3.3.1 The Bessel-Wright Translation Operator
				3.3.2 The Dual of the Bessel-Wright Translation Operator
			3.4 Generalized Wavelet Transform
				3.4.1 Preliminaries
				3.4.2 The Bessel-Wright Wavelet
			3.5 The Heat Kernel
			3.6 The Wave Kernel
		References
	On a Method of Solving Integral Equation of Carleman Type on the Pair of Segments
		References
	Transmutation Operators Boundary Value Problems
		1 Introduction
		2 Materials and Methods
			2.1 The Finite Integral Transforms Technique
				2.1.1 Sturm–Liouville Problem with Dirichlet Boundary Conditions
				2.1.2 Sturm–Liouville Problem with Neumann Boundary Conditions
				2.1.3 Sturm–Liouville Mixed Boundary Value Problem
				2.1.4 Sturm–Liouville Problem with Dirichlet Boundary Conditions on Composite Real Semi-Axis
			2.2 Reflection Method
				2.2.1 Non-local Boundary Value Problem on the Strip
				2.2.2 Boundary Value Problem with Inner Boundary Conditions in a Strip
			2.3 The Fourier Transform Technique
			2.4 Neumann Series Technique
				2.4.1 Solution of the Laplace Equation with Non-local Boundary Conditions in the Strip
				2.4.2 Solution of the Laplace Equation with Generalized Non-local Boundary Conditions in a Strip
		3 Results
		4 Conclusions
		References
	Solution of Inverse Problems for Differential Operators with Delay
		1 Introduction
		2 Auxiliary Propositions
		3 Solution of the Inverse Problem
		References
Part III Transmutations for Partial and Fractional Differential Equations
	Transmutations of the Composed Erdélyi-Kober Fractional Operators and Their Applications
		1 Introduction
		2 The Mellin Integral Transform
		3 Integral Transforms of the Mellin Convolution Type
		4 The Generalized Obrechkoff-Stieltjes Integral Transform
		5 Composed Erdélyi-Kober Fractional Operators and Their Transmutations
		References
	Distributed Order Equations in Banach Spaces with SectorialOperators
		1 Introduction
		2 Nondegenerate Equation at c(0,1]
			2.1 Homogeneous Equation at c(0,1]
			2.2 Inhomogeneous Equation at c(0,1]
		3 Nondegenerate Equation at c>1
			3.1 Homogeneous Equation at c>1
			3.2 Inhomogeneous Equation at c>1
		4 A Class of Initial Boundary Value Problems
		5 Degenerate Distributed Order Equation
			5.1 The Case c(0,1]
			5.2 The Case c(1,2)
		6 Applications to Boundary Value Problems
		References
	Transformation Operators for Fractional Order Ordinary Differential Equations and Their applications
		1 Introduction
		2 Similarity of Fractional Order Ordinary Differential Operators
		3 Similarity of Volterra Operators
		4 Triangular Transformation Operators
			4.1 Sufficient Conditions for Existence of Transformation Operators
			4.2 Necessary Conditions
		5 Uniqueness Results
			5.1 Fractional Order Equations
			5.2 First Order Systems of Ordinary Equations
		6 Completeness of Root Functions of BVPs for Fractional Order Ordinary Differential Equations
		References
	Strong Solutions of Semilinear Equations with Lower Fractional Derivatives
		1 Introduction
		2 Equations Solved with Respect to the Highest Derivative
			2.1 Linear Equation
			2.2 Semilinear Equation
		3 Degenerate Equations
			3.1 Degenerate Semilinear Equation
			3.2 Degenerate Multi-Term Linear Equation
		4 Application
		References
	Mean Value Theorems and Properties of Solutions of Linear Differential Equations
		1 Introduction
		2 Accompanying Distributions
		3 Accompanying Distributions for Singular Operators
		4 Some Examples of Applying the Method
		5 Mean Value Formula for a Two-Dimensional Hyperbolic Equation
		References
	Transmutations for Multi-Term Fractional Operators
		1 Introduction
		2 Fractional Differentiation
		3 Auxiliary Assertions
		4 Transmutation Operator
		5 Application
		References
	Fractional Bessel Integrals and Derivatives on Semi-axes
		1 Introduction
		2 Definitions
			2.1 Special Functions and Integral Transforms
			2.2 Integral Transforms
			2.3 Fractional Bessel Integrals and Derivatives on Semi-axes
				2.3.1 Basic Properties of the Fractional Bessel Integrals on Semi-axes
		3 Factorisation
		4 Resolvent for Fractional Powers of the Bessel Differential Operator
		5 Integral Transforms
			5.1 The Mellin Transform
			5.2 The Hankel Transform
			5.3 The Meijer Transform
			5.4 Generalized Whittaker Transform
		References
	The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay
		1 Introduction
		2 Nonlinear Vibrations of Suspension Bridges and the Method of Multiple Time Scales
			2.1 Nonlinear Undamped Vibrations of Suspension Bridges
			2.2 Nonlinear Damped Free Vibrations of Suspension Bridges
			2.3 Correlation with Experiment
		3 Conclusion
		Appendix
		References
	Boundary Value Problem with Integral Condition for the Mixed Type Equation with a Singular Coefficient
		1 Introduction
		2 Uniqueness
		3 Existence
		4 Stability
		References