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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.
List of contents
One Fundamentals of Semi-Group Theory.- 1.0 Introduction.- 1.1 Elements of Semi-Group Theory.- 1.2 Representation Theorems for Semi-Groups of Operators.- 1.3 Resolvent and Characterization of the Generator.- 1.4 Dual Semi-Groups.- 1.5 Trigonometric Semi-Groups.- 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.- 2.2 Taylor, Peano, and Riemann Operators Generated by Semi-Groups of Operators.- 2.3 Theorems of Non-optimal Approximation.- 2.4 Applications to Periodic Singular Integrals.- 2.5 Approximation Theorems for Resolvent Operators.- 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.- 3.3 Lorentz Spaces and Convexity Theorems.- 3.4 Intermediate Spaces of X and D(Ar).- 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.- 4.2 The Hilbert Transform and the Cauchy-Poisson Singular Integral.- 4.3 The Weierstrass Integral on Euclidean n-Space.- 4.4 Notes and Remarks.
