A retrospective on 60 years of approximation theory and associated fields
Paul L. Butzer
In collecting material for an autobiographical statement, I realized that had been privileged to experience and participate in the emergence evolution of approximation theory across some 60 years. As a student then as professional, observed several strands development within this exciting enterprise. son German emigré family, England Canada, was schooled on two continents, comfortable with three languages, exposed world cultural contrasts. my studies at Toronto (1948–1951), expatriate lecturers who worked British Russian pioneers Fourier analysis approximation. The Cambridge lecturing tutoring style permeated time also lifelong impact classroom presence above all, challenging nurturing relationship generations students. After first position McGill University, spent Paris 2 years Mainz, where very productive interactions US mathematicians. semester stays Freiburg Würzburg, took up Aachen University Technology 1958, when mathematics department expanding its program mission. promotion chair professor 1962, research diversifying training doctoral Convinced importance international collaborative scholarship, organized symposium Oberwolfach Conference Center August 1963. It was, perhaps unexpectedly, successful followed by seven further conferences 20 these symposia—with Béla Szőkefalvi-Nagy co-organizer from fourth onwards—drew about 250 different experts 24 countries including Hungary, Bulgaria, Poland, Roumania eventually Russia, roster participants almost representing Who's Who associated fields such harmonic analysis, functional operator theory, integral transform orthogonal polynomials, interpolation, special functions, divergent series. scope contacts so facilitated served weaken national controls over research, favoring new clusters specialists, partially overlapping interests, all thriving heightened interchange ideas, methods goals. symposia trajectories drew constant stream eminent visitors Aachen, reinforcing exposure our students best patently model quickly adopted Eastern Europe (after 1969) USA 1963/1973), contributing rapid growth popularity fields, complemented appearance journals. vitality overwhelmed negativism Bourbakist critique, but sometimes diluted loose converted narrow cliques, prone citing only each other, more concerned generalizing generalizations than addressing fresh problems or exploring applications. traditional links physics have atrophied, be replaced bridges communications engineering, own case via Walsh functions signal processing (see Section 5). There are, example, intriguing applications numerical ergodic probability well combinatorial number Sections 6 7). Mathematics, particular, flourishes free open exchange requires intensive work begins patient mentoring upcoming
Journal of Approximation Theory