for example, the so-called Lp approximation, the Bernstein approxima- tion problem (approximation on the real line by certain entire functions), and the highly interesting studies of J. L. WALSH on approximation in the complex plane. I would like to extend sincere thanks to Professor L. COLLATZ for his many encouragements for the writing of this book. Thanks are equally due to Springer-Verlag for their ready agreement to my wishes, and for the excellent and competent composition of the book. In addition, I would like to thank Dr. W. KRABS, Dr. A. -G. MEYER and D. SCHWEDT for their very careful reading of the manuscript. Hamburg, March 1964 GUNTER MEINARDUS Preface to the English Edition This English edition was translated by Dr. LARRY SCHUMAKER, Mathematics Research Center, United States Army, The University of Wisconsin, Madison, from a supplemented version of the German edition. Apart from a number of minor additions and corrections and a few new proofs (e. g. , the new proof of JACKSON’S Theorem), it differs in detail from the first edition by the inclusion of a discussion of new work on comparison theorems in the case of so-called regular Haar systems ( 6) and on Segment Approximation ( 11). I want to thank the many readers who provided comments and helpful suggestions. My special thanks are due to the translator, to Springer-Verlag for their ready compliance with all my wishes, to Mr.