My father was increasingly concerned, as early as 1929, about the wave of Nazism sweeping Germany that he believed would lead to another world war. In 1936, my mother engaged an educated man who had spent a year in a concentration camp, to work in our garden. That same year I caused a fuss at my elementary school when the Nazi flag was ceremonially hoisted for a parade, and I ran home shouting that I did not want to take part in such nonsense. "Our sons will never join the Hitler-Jugend (the Hitler Youth organization)", our father always told our mother. "How can you avoid it - since it is compulsory at age ten?" our mother replied.

... the Benedictine Priory at Ealing, and the Collegiate in Bournemouth, a city to which Loewy Engineering moved when London became unsafe during the Nazi bombing raids. It was my fifth school in three countries, each with a different system, within just 5 years.

Convinced of the importance of international and collaborative scholarship, I organized a first international symposium at the Oberwolfach Conference Centre in August 1963. It was, perhaps unexpectedly, successful and was followed by seven further conferences across 20 years. In all, these Oberwolfach symposia - with Béla Szokefalvi-Nagy as co-organizer from the fourth onwards - drew about 250 different experts from 24 countries including Hungary, Bulgaria, Poland, Romania and eventually Russia, the roster of participants almost representing a Who's Who in approximation theory and associated fields such as harmonic analysis, functional analysis and operator theory, integral transform theory, orthogonal polynomials, interpolation, special functions, divergent series. The scope of the contacts so facilitated served to weaken national controls over research, favouring the emergence of new clusters of specialists, with partially overlapping interests, all thriving with the heightened interchange of ideas, methods and goals.

By 1971 Paul's school of approximation theory was well established in Aachen, and in that year the very well-known book 'Fourier Analysis and Approximation', written jointly by Paul and former student R J Nessel, appeared. This ground-breaking project was the first book to emphasize the connections between Fourier analysis and approximation theory. Work in approximation theory continued, with topics such as best trigonometric approximation (extending the Jackson-Bernstein theory), and best approximation by algebraic polynomials, an area where some of Paul's finest results are to be found. Paul's interest in probability theory led him in another direction, beginning in about 1975, to study the central limit theorem and the rates associated with its basic convergence theorem. This problem was, of course, well-suited to Paul's expertise in approximation theory and Fourier analysis. Let us turn to Paul's involvement with signal theory, which began in about 1970 when he became interested in dyadic analysis, particularly the problem of defining a satisfactory derivative in that setting. A few years later, Paul's interests in the sampling theory of band-limited, and of not-necessarily band-limited, functions were awakened by a group of engineers at Aachen; applications of the theory have never been far from his thinking in this area. Sampling theory now became a major activity at the Lehrstuhl A and prompted a wide ranging programme of research involving colleagues, students and their doctoral dissertations, and conferences large and small.

... emphasizes the various semi-groups which pervade approximation theory. The most prominent is the group of translations, but others are integral transforms associated with the names of Abel, Poisson, Gauss and Weierstrass. This book contained the first treatment in book form of the impressive work done on intermediate spaces from the late 1950's on, which was inspired by the classical interpolation (or convexity) theorems of M Riesz-Thorin-Stein and Marcinkiewicz.

This book gives a systematic presentation of the theory of semi-groups of bounded linear operators in a Banach space and of their applications in theoretical questions of approximation. The material in it is based on the contributions of the authors and of their co-workers. ... This book assembles in one place the material from a large number of papers and is of interest to workers in different branches of mathematics. The effect of this monograph will be that semi-group theory, hitherto considered only as a chapter of functional analysis, will become a principal tool of mathematical analysis. Moreover, it has already become indispensable in classical approximation theory, in the study of the initial and boundary behaviour of solutions of partial differential equations and in the theory of singular integrals, because of the new results obtained by the authors in these areas. The systematic treatment and the clarity of the proofs make it possible for this work to be used as a textbook for graduate students.

Approximately half of this volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view. This parallel treatment easily lends itself to an understanding of abstract harmonic analysis; the underlying classical theory is therefore presented in a form that is directed towards the case of arbitrary locally compact abelian groups. The second half is concerned with the concepts making up the fundamental operation of Fourier analysis, namely convolution. Thus, the leitmotiv of the approximation-theoretic part is the theory of convolution integrals, the 'smoothing' of functions by such, and the study of the corresponding degree of approximation. Special as this approach may seem, it not only embraces many of the topics of the classical theory but also leads to significant new results, e.g., on summation processes of Fourier series, conjugate functions, fractional integration and differentiation, limiting behaviour of solutions of partial differential equations, and saturation theory.

The style is very leisurely, too much so at times for my taste, but I would have undoubtedly appreciated it when a student. The notes and remarks are not only interesting, they are quite helpful. A list of symbols is given. This should be given in all books, but authors often do not consider the reader's problems and so omit the extra work required to include this list. Butzer and Nessel and the publishers have been very concerned to ease the problems of the student and are to be commended.

The theme of this monograph is the relation between certain "moment'' problems and various forms of the Riesz representation theorem.

A topic of one master's thesis of 1968 was the mathematical work and lives of the local mathematicians Dirichlet, Prym, and Hamel. Together with R J Nessel and E L Stark we carried out research projects on the life and work of Eduard Helly, with François Jongmans on that of Eugène Catalan and Pafnuty Chebyshev, and with Manfred Jansen and Hubert Zilles on the genealogy and work of Lejeune Dirichlet.

Written by experts from diverse fields, the book aims to communicate both historical perspectives and modern ideas across disciplinary boundaries. Careful editing has preserved coherence in the exposition. The book is of a rare genre, and it is highly successful.

Paul L Butzer presents a broad but un-specialized survey of scholarship in mathematics and astronomy during Carolingian times, in the explicit context of the older sources available to early medieval writers and in comparison with parallel developments in the Byzantine and Islamic worlds.