A retrospective on 60 years of approximation theory and associated fields

AbstractIn collecting material for an autobiographical statement, I realized that I had been privileged to experience and participate in the emergence and evolution of approximation theory across some 60 years. As a student and then as a professional, I had observed the several strands of development within this exciting enterprise. As the son of a German emigré family, in England and Canada, I was schooled on two continents, comfortable with three languages, and exposed to a world of cultural contrasts. In my studies at Toronto (1948–1951), I was exposed to expatriate lecturers who had worked with the British and Russian pioneers in Fourier analysis and approximation. The Cambridge lecturing and tutoring style that permeated Toronto at the time also had a lifelong impact on my classroom presence and above all, my challenging and nurturing relationship with several generations of students.After a first position at McGill University, I spent some time in Paris and then 2 years in Mainz, where I had very productive interactions with German and US mathematicians. After semester stays in Freiburg and Würzburg, I took up a position at the Aachen University of Technology in 1958, at a time when the mathematics department was expanding its program and mission. After my promotion to chair professor in 1962, my research was diversifying and I was training my first doctoral students.Convinced of the importance of international and collaborative scholarship, I organized a first international symposium at the Oberwolfach Conference Center 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 Szőkefalvi-Nagy as co-organizer from the fourth onwards—drew about 250 different experts from 24 countries including Hungary, Bulgaria, Poland, Roumania 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, favoring the emergence of new clusters of specialists, with partially overlapping interests, all thriving with the heightened interchange of ideas, methods and goals. The symposia and new research trajectories drew a constant stream of eminent visitors to Aachen, reinforcing the exposure of our students to the best of a patently international enterprise.The Oberwolfach model was quickly adopted in Eastern Europe (after 1969) and the USA (after 1963/1973), contributing to the rapid growth and popularity of approximation theory and associated fields, that was complemented by the appearance of new journals. The new vitality of the fields overwhelmed the negativism of the Bourbakist critique, but was sometimes diluted when loose clusters converted to narrow cliques, prone to citing only each other, and more concerned with generalizing generalizations than addressing fresh problems or exploring productive applications. The traditional links to physics have atrophied, to be replaced by bridges to communications engineering, in my own case via Walsh functions and signal processing (see Section 5). There also are, for example, intriguing applications to functional analysis, numerical analysis, ergodic theory, probability theory, as well as combinatorial number theory (see Sections 6 and 7). Mathematics, and approximation theory in particular, flourishes best with a free and open exchange of ideas, and requires intensive collaborative work that begins with the training and patient mentoring of upcoming students