**Paul Butzer**'s parents, Anton Paul Butzer and Wihelmine Hansen, met while students at RWTH (Rheinisch Westfälische Technische Hochschule) in Aachen. Anton, originally from Düsseldorf, was studying mechanical engineering while Wihelmine, a native of Aachen, was the first woman to study mathematics at RWTH. Anton and Wihelmine Butzer married in 1925 and moved to Mülheim an der Ruhr where Anton worked as an engineer. Soon after Paul was born, the political situation in Germany began to worry the family, as Butzer describes in his autobiographical article [5]:-

My father was increasingly concerned, as early as1929, about the wave of Nazism sweeping Germany that he believed would lead to another world war. In1936, 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.

In fact Anton Butzer took drastic steps to avoid his family becoming swept up in the Nazi wave. His opposition to the Nazis was already leading to increasingly serious repercussions by spring 1937 when he left Germany to attend an engineering exhibition in Arnhem, in the Netherlands. He had purchased a return train ticket but had no intention of returning. Once in Arnhem he contacted Ludwig Loewy, a friend who had emigrated to England three years earlier. Loewy sent money to Anton in Arnhem who was then able to travel to England. He joined Loewy's engineering firm in London but, fearing that Paul and his younger brother Karl (about 30 months old at that time) would suffer for their father's actions, they had been sent to Gemmenich, a town in Belgium just across the border from Aachen. There they lived at a convent attached to a girl's boarding school while Wihelmine managed to join her husband in London. Once their parents were settled, Paul and Karl were able to join them and, by the autumn of 1937, Paul was studying at the Jesuit Wimbledon College. However, frequent moves followed, and he next attended [5]:-

... 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 just5years.

After a six week internment on the Isle of Man in 1940, along with his mother and brother, Butzer was able to continue schooling until the family were sent to New York where Loewy's engineering firm were opening a new branch. They sailed in January 1941 but after reaching Saint John's, New Brunswick, as German nationals, they were not allowed to enter the United States. Butzer's father got a position with a company in Montreal and in February 1941 he entered Loyola High School. After graduating, he entered Loyola College (now Concordia University) in 1944. After a year he decided to major in mathematics and was awarded a B.Sc. with Honours in 1948. He then entered the University of Toronto where he took courses for a Master's degree. Among his lecturers we mention Donald Coxeter, who gave courses on non-Euclidean geometry and number theory, and Bill Tutte who gave a topology course. Butzer received an M.A. in 1949, being awarded the Sir Joseph Flavelle Prize the best performance. He then continued studying at Toronto for his doctorate, supervised by George Lorentz. As this time Lorentz was writing his book *Berstein polynomials* so work in this area was a natural choice as a topic for Butzer.

In 1951 Butzer was awarded a Ph.D. for his thesis *On Bernstein Polynomials* in which he studied classical Bernstein polynomials and their variant for integrable functions, namely Kantorovich polynomials. In particular he studied their convergence in a number of different spaces. In 1952 he was appointed as a lecturer at McGill University in Montreal and promoted to assistant professor in the following year. In 1954 the International Congress of Mathematicians was held in Amsterdam from 2 September to 9 September. This European experience made him keen to spend research leave in Europe and he spent the year 1955-56 first spending a short time in Paris then spending most of the year in Mainz. He spent a second year 1956-57 at Mainz as a visiting professor. At this stage he decided to resign his position at McGill University and spent a semester at Freiburg University with Wilhelm Süss. Of course he was not qualified for a proper position at a German university so, while at Freiburg, he submitted material for his habilitation. After teaching for the summer semester of 1958 at Würzburg University, he began teaching at RWTH in Aachen.

In 1962 he was offered a chair at Groningen, in the Netherlands, but this prompted RWTH to also offer Butzer a chair. He accepted Aachen's offer, heading Lehrstuhl A für Mathematik and building a strong school of mathematics there. He explains in [5] how he organised a remarkably successful series of conferences:-

Convinced of the importance of international and collaborative scholarship, I organized a first international symposium at the Oberwolfach Conference Centre in August1963. It was, perhaps unexpectedly, successful and was followed by seven further conferences across20years. In all, these Oberwolfach symposia - with Béla Szokefalvi-Nagy as co-organizer from the fourth onwards - drew about250different experts from24countries 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.

In [10] Higgins gives a summary of Butzer's research contributions:-

By1971Paul'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 about1975, 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 about1970when 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.

Let us look briefly at the books that Butzer has published, with the help of some of his students. First there is the monograph (with Hubert Berens) *Semi-groups of operators and approximation* (1967). Richard Askey explains that the book [1]:-

... 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 late1950's on, which was inspired by the classical interpolation(or convexity)theorems of M Riesz-Thorin-Stein and Marcinkiewicz.

D Pascali writes in a review:-

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.

Following rapidly in quick succession there is (with Walter Trebels) *Hilberttransformation, gebrochene Integration und Differentiation* Ⓣ (1968), and (with Karl Scherer) *Approximationsprozesse und Intepolationsmethoden* Ⓣ (1968). Two years later Butzer published (with Rolf J Nessel) *Fourier analysis and approximation. Volume 1: One-dimensional theory* (1970). The authors write in the Preface:-

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.

Richard Askey writes [1]:-

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.

Finally we mention (with W Oberdörster) *Darstellungssätze für beschränkte lineare Funktionale im Zusammenhang mit Hausdorff-, Stieltjes- und Hamburger-Momentenproblemen* Ⓣ (1975):-

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

We must, particularly in a History of Mathematics Archive, note Butzer's interest in the history of mathematics [5]:-

A topic of one master's thesis of1968was 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.

In 1981 Butzer edited, in collaboration with F Fehér, *E B Christoffel: The Influence of His Work on Mathematics and the Physical Sciences*. As well as editing the book, Butzer contributed an excellent article on the life and work of Christoffel. The book received outstanding reviews; for example James Rovnyak [14] writes:-

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.

Similarly, in 1993 Butzer edited, in collaboration with Dietrich Lohrmann, *Science in Western and Eastern Civilization in Carolingian Times*. As well as commenting that the book is "well-edited", H L L Busard writes [4]:-

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.

Finally we note that Butzer has received many honours for his outstanding contributions. He has been elected to the Société Royale des Sciences de Liège, the Académie Royale de Belgique and honorary membership of the Mathematische Gesellschaft in Hamburg. He has been awarded honorary degrees from the University of Liège, the University of York in England, and Timișoara University. The conference 'Approximation Theory and Signal Analysis', organised by the Institute of Biomathematics and Biometry at the Helmholtz Center Munich, was held in his honour on the occasion of his 80^{th} birthday in 2009 at the Hotel Bayerischer Hof, Lindau (Lake Constance), Germany.

**Article by:** *J J O'Connor* and *E F Robertson*