Advancements in Complex Analysis: From Theory to Practice

Preface
Contents
A Theory on Non-Constant Frequency Decompositionsand Applications
	1 Introduction
	2 Mono-component Function Theory
		2.1 Mono-component and IF
		2.2 The Inner Function Type Mono-Components
		2.3 The Bedrosian Type Mono-Components
		2.4 The Non-Bedrosian Type Mono-Components: The Starlike and Boundary Starlike Type
	3 Adaptive Fourier Approximations
		3.1 Mono-component Decomposition of Signals in General
		3.2 One Dimensional Core-Adaptive Fourier Decomposition (Core-AFD) and Its Variations
		3.3 Unwinding AFD (UWAFD)
		3.4 Cyclic AFD for n-Best Rational Approximation
		3.5 Pre-Orthogonal Adaptive Fourier Decomposition (POAFD) for Reproducing Kernel Hilbert Spaces
	4 Related Studies and Applications
		4.1 Aspects in Relation to Beurling–Lax Shift-Invariant Subspaces
		4.2 Extra-Strong Uncertainty Principle
		4.3 The Dirac-Type Time-Frequency Distributions Based on Mono-component Decompositions
		4.4 Higher Dimensional AFDs
		4.5 Fourier Spectrum Characterization of Hardy Spaces: Analytic Signals Revised
		4.6 Hilbert Transforms as Singular Integral Operators: Analytic Signals Revised
		4.7 Applications
	References
One-Component Inner Functions II
	1 Introduction
	2 Main Tools
	3 Splitting Off Factors
	4 Composition of One-Component Inner Functions
	References
Biholomorphic Cryptosystems
	1 Introduction
	2 Biholomorphic Codes
	3 Convergence Properties of Biholomorphic Codes
	4 Dynamic Properties of Biholomorphic Codes
	5 Biholomorphic Cryptosystems
	6 Dynamics of Biholomorphic Cryptosystems
	References
Third-Order Fermionic and Fourth-Order Bosonic Operators
	1 Introduction
	2 Preliminaries
		2.1 Clifford Algebra
		2.2 Irreducible Representations of the Spin Group
			Spinor Representation of Spin(m)
			Homogeneous Harmonic Polynomials on Hk(Rm,C)
			Homogeneous Monogenic Polynomials on Clm
	3 Construction and Conformal Invariance
		3.1 3rd-Order Higher Spin Operator D3
			Detailed Proof of Proposition 1
		3.2 4th-Order Higher Spin Operator D4
			Detailed Proof of Proposition 2
	4 Fundamental Solutions and Intertwining Operators
	5 Connection with Lower Order Conformally Invariant Operators
	6 Ellipticity
		6.1 Ellipticity for 3rd-Order Higher Spin Operator D3
		6.2 Ellipticity for 4th-Order Higher Spin Operator D4
	References
Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan
	1 Introduction
	2 From Weierstrass and Runge to Mergelyan
	3 Approximation on Unbounded sets in Riemann Surfaces
	4 Mergelyan's Theorem for Cr Functions on Riemann Surfaces
	5 The Oka–Weil Theorem and Its Generalizations
	6 Mergelyan's Theorem in Higher Dimensions
		6.1 Approximation on Totally Real Submanifolds and Admissible Sets
		6.2 Approximation on Strongly Pseudoconvex Domains and on Strongly Admissible Sets
		6.3 Mergelyan Approximation in L2-Spaces
		6.4 Carleman Approximation in Several Variables
	7 Approximation of Manifold-Valued Maps
		7.1 Runge Theorem for Maps from Stein Spaces to Oka Manifolds
		7.2 Mergelyan Theorem for Manifold-Valued Maps
		7.3 Carleman and Arakelian Theorems for Manifold-Valued Maps
	8 Weighted Approximation in L2 Spaces
	9 Appendix: Whitney's Extension Theorem
	References
A Potapov-Type Approach to a Truncated Matricial Stieltjes-Type Power Moment Problem
	1 Introduction and Preliminaries
	2 On the Solvability of Matricial Power Moment Problems
	3 Some Classes of Holomorphic Matrix-Valued Functions
	4 On the Equivalence of the Stieltjes Moment Problem to a System of Two Fundamental Matrix Inequalities of Potapov Type
	5 Some Considerations on Block Hankel Matrices
	6 Dubovoj Subspaces and Associated Generalized Inverses of Matrices
	7 Construction of a Pair of Coupled J̃q-Inner 2q2q Matrix Polynomials
	8 Stieltjes Pairs of Meromorphic Matrix-Valued Functions
	9 The Class W̃J̃q,α
	10 On the Class WJ̃q,α Under the View of Linear Fractional Transformations
	11 On the Solutions of the Schur Complement Matrix Inequalities
	12 On a Closer Analysis of the Range Conditions in Proposition 4.10
	13 On a First Description of the Set S0,q;[α,∞)(sj)j=02n+1,≤
	14 A Pair of Subspaces of Cq Which Describes the Degeneracy of the Moment Problem MP[[α,∞);(sj)j=02n+1,≤]
	15 A Further Parametrization of the Solution Set of the Truncated Matricial Stieltjes Moment Problem in the Degenerate But Not Completely Degenerate Case
	16 The Completely Degenerate Case
	17 A Particular Generalized Inverse of a Complex Matrix
	References
Formulas and Inequalities for Some Special Functionsof a Complex Variable
	1 Introduction
	2 An Identity for Three Gauss Hypergeometric Functions
	3 General Convolution Inequalities
	4 Inequalities for Bessel Functions
	5 Inequalities for Laguerre and Hermite Polynomials
	6 Inequalities for Whittaker Functions
	References
On the Means of the Non-trivial Zeros of the Riemann Zeta Function
	1 Introduction and Summary of the Results
	2 Proofs
		2.1 Proof of (3)
		2.2 Proof of (4)
		2.3 Proof of (5)
		2.4 Proof of Theorem 1
	References
Minimal Kernels and Compact Analytic Objects in Complex Surfaces
	1 Introduction
	2 The Minimal Kernel and Its Slices
	3 Compact Complex Curves
		3.1 The Neighborhood of a Compact Curve
			Grauert Criterium
			Positive Curves
			Ueda's Paper
			Curves Near C
		3.2 Propagation of Compact Curves
			Nishino's Paper
			Ohsawa's Paper and Its Generalization
	4 Weakly Complete Surfaces
		4.1 Examples and Remarks
		4.2 Classification Results
		4.3 Coronae of Dimension 2
		4.4 Brunella's Example
	5 Minimal Kernels and the Structure of Complex Manifolds
		5.1 Complex Surfaces with a Smooth Exhaustion
		5.2 The Singular Locus of an Admissible Class
		5.3 A Levi Problem
	References
On the Automorphic Group of an Entire Function
	1 Some Background and the Contribution of Tatsujirô Shimizu
	2 The Weierstrass Representation of the Automorphic Group of an Entire Function, and the Extra Properties in the Case of a Finite Order
	3 Conclusions from Proposition 3 in Case We Have No Monodromy
	4 The Cycle Relation and the Chain Relation in the General Case
	5 Examples (Mostly the Exponential Function) and the Role Played by the Assumption That We Have Some Summation Method for the Infinite Series: n=1∞Qλn(wϕ0n(z)), for the Reconstruction of f from Aut(f)
	6 Reconstruction Formulas for f(z) and for f'(z) in Terms of Approximating Automorphic Functions: Relations Between the Groups Aut(f) and Autz(g(w,z))
	7 The Function g(w,z)-g(0,z) Is Determined by the Negative Moments of the Elements in Aut(f(z))
	8 An Infinite Product Representation of f'(w)
	9 Common Zeros of the Reciprocals of Almost All the Automorphic Functions
	10 Sums of the Derivatives of the Automorphic Functions
	11 An Application of Jensen's Theorem to the Automorphic Group of an Entire Function
	12 A Computation of the Integral 12π02πlog|f(reiθ)-f(z)|dθ
	13 The Product of the Automorphic Functions
	14 Consequences to Aut(f) That Follow from the Classical Theory of Entire Functions
	15 The Relations Between Scattering Theory and Automorphic Functions
	16 Local Groups
	17 The Sums of the k'th Derivatives of All the Elements of the Automorphic Group Aut(f), for Any fE of Order 0<ρ<12, k=1,2,3,…
	18 The Circular Density of the Orbits of the Automorphic Group Aut(f), for Any fE of Order 0<ρ<12
	19 The Vieta Formulas for Aut(f), fE of Order 0≤ρ<1
	20 Embedding the Automorphic Group Within a Larger Group
	21 Relations Between the Construction of the Direct System of the Automorphic Groups and Weierstrass Products
	22 Continuity Properties of the Automorphic Groups
	23 Amenability of the Automorphic Group
	References
Integral Representations in Complex Analysis: From Classical Results to Recent Developments
	1 Classical Results
		1.1 Results Up to the 1940s
		1.2 Leray's New Kernel Construction
		1.3 Kernels for Strictly Pseudoconvex Domains
	2 Beyond Strictly Pseudoconvex Domains: Many Problems
		2.1 L2 Results and Finite Type
		2.2 The Obstruction to Holomorphic Kernels
		2.3 Partial Results for Convex Domains and in Dimension Two
		2.4 Speculation on Some Possible Approaches
	3 A New Kernel Approach
		3.1 Motivation: The Basic L2 A-priori Estimate
		3.2 A New Kernel
		3.3 Pointwise A-priori Estimates
		3.4 Outlook: A-priori Hölder Estimates
		3.5 Some Conjectures
	References
On the Riemann Zeta Function and Gaussian Multiplicative Chaos
	1 Background: Classical Results on Statistics of ζ
		1.1 Towards Functional Statistics of ζ: The Easy Case of σ>1
		1.2 Bohr, Jessen, and Bagchi: The Case of σ>1/2
		1.3 Selberg: Pointwise Statistics in the Case of σ=1/2
	2 Log-Correlated Fields
		2.1 Emergence of Log-Correlated Field: Heuristics and Facts
		2.2 Log-Correlated Gaussian Fields
	3 Multiplicative Chaos
		3.1 Gaussian Multiplicative Chaos Measures
		3.2 Critical and Supercritical Chaos
		3.3 Complex Chaos
	4 Riemann Zeta and Multiplicative Chaos
		4.1 Some Ingredients of the Proof of Part (i) of Thm 4.1
		4.2 Some Ingredients of Part (ii) of Thm 4.1
	5 The Mesoscopic Scale: ζ Meets Random Matrices
		5.1 The Montgomery(-Dyson) Paradigm
		5.2 Rigorous Results on the Mesoscopic Scale
	6 Results and Conjectures for Statistics of |ζ(1/2+it)|β
		6.1 The Fyodorov–Hiary–Keating Conjecture
		6.2 Multiplicative Chaos as Statistical Limits for Shifts of |ζ(1/2+it)|β?
	References
Some New Aspects in Hypercomplex Analysis
	1 Hyperquaternions
		1.1 Introduction
		1.2 Multiplication in HH
		1.3 Generators
		1.4 Clifford Numbers
		1.5 Clifford Algebras over the Real Numbers
		1.6 Hyperquaternions for Classification of Physics
	2 Analysis on the 3-Sphere—Some Topics
		2.1 Representations of S3
		2.2 Tomographic Methods
	3 Fluid Flow Through Porous Media with the Help of a Quaternionic Operator Calculus
		3.1 Some Basic Fluid Flow Equations
		3.2 A Quaternion Operator System
		3.3 Representation in Terms of (Dα)-Holomorphic Functions
	4 An Adaptive Fast Fourier Type Decomposition
		4.1 Takenaka–Malmquist Systems
	5 Harmonic Conjugates in Weighted Bergman Spaces
	6 On Schwarz Type Formulae
		6.1 Schwarz Integral Formula in the Complex Plane
		6.2 Schwarz Kernel in R4
		6.3 Schwarz Formula for the Ball in R3
	References
Some Connections of Complex Dynamics
	1 Introduction
	2 Background Material in Complex Dynamics
	3 Complex Dynamics and Kleinian Groups
	4 Newton's Method and Other Numerical Methods
		4.1 Genericity of Convergence
		4.2 Finding All the Roots
	5 Hierarchical Ising and Potts Models
	6 Other Connections
	References