George G. Lorentz - Böcker

The works of George G. Lorentz, spanning more than 60 years, have played a significant role in the development and evolution of mathematical analysis. The papers presented in this volume represent a selection of his best works, along with commentary from his students and colleagues.

This "Selecta" contains approximately two thirds of the papers my father wrote from 1932 to 1994. These papers are divided into four fields. The first volume contains the papers on 1) Summability and Number Theory and 2) Interpolation. The second volume contains the fields 3) Real and Functional Analysis and 4) Approximation Theory. Each of these four groups of papers is introduced by a review of the contents and significance, respectively of the impact of these papers. The first volume contains, in addition, an autobiography, a complete list of publications, a list of doctoral students and four unpublished essays on mathematics in general: a) A report on the University of Leningrad b) On the work of the mathematical mind c) Proofs in Mathematics d) About Mathematical books. The report on the University of Leningrad, written in the late '40's, is a unique historical document which is still of current interest for several reasons. It is of interest for professional reasons since it contains a com­ plete description of a mathematics majors' curriculum through his entire course of studies. From it one can see both the changes and invariants of course material as well as the students' course load. Then one can also see the consequences of admittedly extreme political intervention in uni­ versity affairs. Today we use the term "politically correct", but in those times being politically correct was a matter of life and death. Finally, this is a tragedy of human beings caught in the siege of Leningrad.

The present book deals with some basic problems of Approximation Theory: with properties of polynomials and splines, with approximation by poly- mials, splines, linear operators. It also provides the necessary material ab out different function spaces. In some sense, this is a modern version of the corre­ sponding parts of the book of one of us (Lorentz [A-1966]). We have tried to give a complete exposition of the main, basic theorems of the theory, without going into too much detail, treating the most general cases or discussing very special problems. There are essential limitations: this is a book about approximation of functions of one real variable. But Approxima­ tion Theory of functions of several real or of complex variables would require new books. Very little is given on interpolation. But even with these restric­ tions, proofs of some deep and important results, like Korneichuk's theorems about approximation in Lipschitz spaces, could not be included. Another book, with different authors, "Constructive Approximation, Advanced Problems", is in preparation. There is an extensive bibliography, which can be used also as an Author Index: each paper is supplied with references for the page or pages where it has been used.

The present book deals with some basic problems of Approximation Theory: with properties of polynomials and splines, with approximation by poly- mials, splines, linear operators. It also provides the necessary material ab out different function spaces. In some sense, this is a modern version of the corre­ sponding parts of the book of one of us (Lorentz [A-1966]). We have tried to give a complete exposition of the main, basic theorems of the theory, without going into too much detail, treating the most general cases or discussing very special problems. There are essential limitations: this is a book about approximation of functions of one real variable. But Approxima­ tion Theory of functions of several real or of complex variables would require new books. Very little is given on interpolation. But even with these restric­ tions, proofs of some deep and important results, like Korneichuk's theorems about approximation in Lipschitz spaces, could not be included. Another book, with different authors, "Constructive Approximation, Advanced Problems", is in preparation. There is an extensive bibliography, which can be used also as an Author Index: each paper is supplied with references for the page or pages where it has been used.