In recent years important progress has been made in the study of semi-groups of operators from the viewpoint of approximation theory. These advances have primarily been achieved by introducing the theory of intermediate spaces. The applications of the theory not only permit integration of a series of diverse questions from many domains of mathematical analysis but also lead to significant new results on classical approximation theory, on the initial and boundary behavior of solutions of partial differential equations, and on the theory of singular integrals. The aim of this book is to present a systematic treatment of semi­ groups of bounded linear operators on Banach spaces and their connec­ tions with approximation theoretical questions in a more classical setting as well as within the setting of the theory of intermediate spaces. However, no attempt is made to present an exhaustive account of the theory of semi-groups of operators per se, which is the central theme of the monumental treatise by HILLE and PHILLIPS (1957). Neither has it been attempted to give an account of the theory of approximation as such. A number of excellent books on various aspects of the latter theory has appeared in recent years, so for example CHENEY (1966), DAVIS (1963), LORENTZ (1966), MEINARDUS (1964), RICE (1964), SARD (1963). By contrast, the present book is primarily concerned with those aspects of semi-group theory that are connected in some way or other with approximation.

Inhalt

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.