Semi-Groups of Operators and Approximation
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 the
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Herausgegeben von
J.L.Doob . E.Heinz· F.Hirzebruch· E.Hopf. H.Hopf W.Maak· S. Mac Lane . W.Magnus· D.Mumford M. M. Postnikov . F. K. Schmidt· D. S. Scott· K. Stein
C;esch~t~uorende
Herausgeber
B. Eckmann und B. L. van der Waerden
Paul L.Butzer . Hubert Berens
Semi-Groups of Operators and Approximation
Springer-Verlag New York Inc. 1967
Professor Dr. Paul L. Butzer Rheinisch·Westfa.lische Technische Hochschule Aachen
Dr. rer. nat. Hubert Berens Rheinisch-\\lestfalische Technische Hochschule Aachen
Geschaftsfiihrende Herausgeber:
Professor Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich
Professor Dr. B. L. van der Waerden Mathematisches Institut der Universitat Zurich
ISBN-13: 978-3-642-46068-5 e-ISBN-I3: 978-3-642-46066-1 DOl: 10.1007/978-3-642-46066-1 All righ ts reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag
© by Springer-Verlag, Berlin· Heidelberg
1967
Softcover reprint of the hardcover 1st edition 1967 Library of Congress Catalog Card Number 68-II980 Title NO.5 128
Preface 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 semigroups of bounded linear operators on Banach spaces and their connections 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. Special emphasis is placed upon the significance of the relationships between the abstract theory and its various applications. This is, in fact, the original aim of the Springer Grundlehren as suggested by the subtitle of the series. The present book is written for the graduate student as well as for the research mathematician. It can be read by one who is familiar with real variable theory and the elements of functional analysis. To make the exposition self-contained these foundations a
