Semi-Groups of Operators and Approximation

One Fundamentals of Semi-Group Theory.- 1.0 Introduction.- 1.1 Elements of Semi-Group Theory.- 1.1.1 Basic Properties.- 1.1.2 Holomorphic Semi-Groups.- 1.2 Representation Theorems for Semi-Groups of Operators.- 1.2.1 First Exponential Formula.- 1.2.2 General Convergence Theorems.- 1.2.3 Weierstrass Approximation Theorem.- 1.3 Resolvent and Characterization of the Generator.- 1.3.1 Resolvent and Spectrum.- 1.3.2 Hille-Yosida Theorem.- 1.3.3 Translations; Groups of Operators.- 1.4 Dual Semi-Groups.- 1.4.1 Theory.- 1.4.2 Applications.- 1.5 Trigonometric Semi-Groups.- 1.5.1 Classical Results on Fourier Series.- 1.5.2 Fourier’s Problem of the Ring.- 1.5.3 Semi-Groups of Factor Sequence Type.- 1.5.4 Dirichlet’s Problem for the Unit Disk.- 1.6 Notes and Remarks.- Two Approximation Theorems for Semi-Groups of Operators.- 2.0 Introduction.- 2.1 Favard Classes and the Fundamental Approximation Theorems.- 2.1.1 Theory.- 2.1.2 Applications to Theorems of Titchmarsh and Hardy-Littlewood.- 2.2 Taylor, Peano, and Riemann Operators Generated by Semi-Groups of Operators.- 2.2.1 Generalizations of Powers of the Infinitesimal Generator.- 2.2.2 Saturation Theorems.- 2.2.3 Generalized Derivatives of Scalar-valued Functions.- 2.3 Theorems of Non-optimal Approximation.- 2.3.1 Equivalence Theorems for Holomorphic Semi-Groups.- 2.3.2 Lipschitz Classes.- 2.4 Applications to Periodic Singular Integrals.- 2.4.1 The Boundary Behavior of the Solution of Dirichlet’s Problem; Saturation.- 2.4.2 The Boundary Behavior for Dirichlet’s Problem; Non-optimal Approximation.- 2.4.3 Initial Behavior of the Solution of Fourier’s Ring Problem.- 2.5 Approximation Theorems for Resolvent Operators.- 2.5.1 The Basic Theorems.- 2.5.2 Resolvents as Approximation Processes.- 2.6 Laplace Transforms in Connection with a Generalized Heat Equation.- 2.7 Notes and Remarks.- Three Intermediate Spaces and Semi-Groups.- 3.0 Scope of the Chapter.- 3.1 Banach Subspaces of X Generated by Semi-Groups of Operators.- 3.2 Intermediate Spaces and Interpolation.- 3.2.1 Definitions.- 3.2.2 The K- and J-Methods for Generating Intermediate Spaces.- 3.2.3 On the Equivalence of the K- and J-Methods.- 3.2.4 A Theorem of Reiteration.- 3.2.5 Interpolation Theorems.- 3.3 Lorentz Spaces and Convexity Theorems.- 3.3.1 Lorentz Spaces.- 3.3.2 The Theorems of M. Riesz-Thorin and Marcinkiewicz.- 3.4 Intermediate Spaces of X and D(Ar).- 3.4.1 An Equivalence Theorem for the Intermediate Spaces X?, r; q.- 3.4.2 Theorems of Reduction for the Spaces X?, r; q.- 3.4.3 The Spaces X0?, r; ?.- 3.5 Equivalent Characterizations of X?, r; q Generated by Holomorphic Semi-Groups.- 3.6 Notes and Remarks.- Four Applications to Singular Integrals.- 4.0 Orientation.- 4.1 Periodic Functions.- 4.1.1 Generalized Lipschitz Spaces.- 4.1.2 The Singular Integral of Abel-Poisson.- 4.1.3 The Singular Integral of Weierstrass.- 4.2 The Hilbert Transform and the Cauchy-Poisson Singular Integral.- 4.2.1 Foundations on the Fourier Transform.- 4.2.2 The Hilbert Transform.- 4.2.3 The Singular Integral of Cauchy-Poisson.- 4.3 The Weierstrass Integral on Euclidean n-Space.- 4.3.1 Sobolev and Besov Spaces.- 4.3.2 The Gauss-Weierstrass Integral.- 4.4 Notes and Remarks.