Walter Gautschi, Volume 1

Walter Gautschi was born on December 11, 1927 in Basel, Switzerland, together with his twin brother Werner. He attended primary and secondary schools in Basel, graduating in 1947 from the Mathematisch-Naturwissenschaftlichen Gymnasium. He then enrolled at the University of Basel to study mathematics as the primary subject, with physics, physical chemistry, and actuarial mathematics as secondary subjects. In the early 1950s he became an assistant of Professor Alexander M. Ostrowski, obtaining a Ph. D. in 1953 under his supervision with a thesis on graphical integration of ordinary differential equations. He then received a twoyear fellowship for study abroad from the Janggen-Poehn foundation in St. Gallen, of which he spent the first year at the Istituto Nazionale per le Applicazioni del Calcolo in Rome, founded and directed by Mauro Picone, and a second year at the Harvard Computation Laboratory. It was at the Harvard Computation Laboratory where he got his first hands-on experience with electronic computers, programming (in machine code) on Professor Aiken’s MARK III computer. In 1956, under a contract with the American University, he joined the staff of the Computation Laboratory at the National Bureau of Standards in Washington, D. C. (now the National Institute of Standards and Technology). There, his major project was the preparation of two chapters of the Handbook of Mathematical Functions edited by Milton Abramowitz and Irene A. Stegun. Abramowitz introduced Walter to the work of J. C. P. Miller on backward recurrence, which became one of the early areas of emphasis in Walter’s research. Because of employment difficulties related to Walter’s Swiss citizenship, he had to leave the Bureau in 1959 and he joined Alston Householder’s Mathematics Panel at the Oak Ridge National Laboratory. Through contacts with chemists at the laboratory, he became interested in the numerical aspects of Gaussian quadrature and orthogonal polynomials, which was to become one of the principal areas of Walter’s research contributions. During the four years at the Oak Ridge laboratory he was twice invited to lecture at the Michigan University Engineering Summer Conferences then organized by Robert C. F. Bartels.

I have worked in a number of different areas of (mostly computational) mathematics. They are organized here in thirteen sections. For the sake of brevity, when referring to joint papers, coauthors are not identified explicitly.

A theme running through Gautschi’s work is numerical conditioning. His many papers on this topic fall broadly into two categories: those on conditioning of Vandermonde matrices and those on conditioning of polynomials.

The collection of papers by Walter Gautschi dealing with special functions has, of course, connections with other sections in these volumes. First, we have to mention the article [GA29], which is included in Section 14 dedicated to difference equations, where the conditioning of three-term recurrence relations is analyzed and methods of computation using recurrence relations are developed; for a more recent review, see [GA150]. Recurrence relations are basic tools for computing special functions, particularly functions of hypergeometric type. In [GA29], also the relation between the existence of a minimal solution for the recurrence and the convergence of the associated continued fraction is discussed. Reference [GA29] is a pioneering and influential paper in the field of special functions, and it is a highly cited paper (316 citations as of now).

In the papers collected here, Walter Gautschi makes vital contributions to the theory of interpolation and approximation. He considers attenuation factors in practical Fourier analysis, Padé approximants associated with Hamburger series, the convergence behavior of continued fractions with real elements, moment-preserving spline approximations, and the convergence of extended Lagrange interpolation. Further, he uses numerical computations to examine the validity of mathematical conjectures regarding zeros of Jacobi polynomials and weighted Newton–Cotes quadrature formulae.