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Remembering Walter Rudin (1921–2010) Remembering Walter Rudin (1921–2010) Alexander Nagel and Edgar Lee Stout, Coordinating Editors alter Rudin, Vilas Professor Emeri- 1945. He entered Duke Uni- tus at the University of Wisconsin- versity, obtaining a B.A. in Madison, died on May 20, 2010, at 1947 and a Ph.D. in math- his home in Madison after a long ematics in 1949. He was battle with Parkinson’s disease. He a C. L. E. Moore Instructor W at the Massachusetts In- was born in Vienna on May 2, 1921. The Rudins were a well-established Jewish stitute of Technology and family which began its rise to prominence in began teaching at the Uni- the first third of the nineteenth century. By the versity of Rochester in 1830s, Walter’s great-grandfather, Aron Pollak, 1952. had built a factory to manufacture matches; he While on leave visiting also became known for his charitable activities, Yale in 1958, Rudin re- including the construction of a residence hall ceived a call from R. H. where seventy-five needy students at the Technical Bing at the University of University in Vienna could live without paying rent. Wisconsin-Madison, ask- As a result, Aron was knighted by Emperor Franz ing if he would be Joseph in 1869 and took the name Aron Ritter interested in teaching sum- Pollak von Rudin. The Rudin family prospered, and mer school. Rudin said Walter’s father, Robert, was a factory owner and that, since he had a Sloan Fellowship, he wasn’t inter- Photograph courtesy of Mary Ellen Rudin. electrical engineer, with a particular interest in Walter Rudin and sister, Vera, in ested in summer teaching. sound recording and radio technology. He married Vienna. Walter’s mother, Natalie (Natasza) Adlersberg, in Then, as he writes in his autobiography, As I Remember It, “my brain slipped 1920. Walter’s sister, Vera, was born in 1925. out of gear but my tongue kept on talking and I After the Anschluss in 1938, the situation for heard it say ‘but how about a real job?’ ” As a result, Austrian Jews became impossible, and the Rudin Walter Rudin joined the Department of Mathemat- family left Vienna. Walter served in the British ics at UW-Madison in 1959, where he remained Army and Navy during the Second World War, and until his retirement as Vilas Professor in 1991. rejoined his parents and sister in New York in late He and his wife, the distinguished mathematician Mary Ellen (Estill) Rudin, were popular teachers at Alexander Nagel is emeritus professor of mathematics at the University of Wisconsin-Madison. His email address is both the undergraduate and graduate level and [email protected]. served as mentors for many graduate students. Edgar Lee Stout is emeritus professor of mathematics at the They lived in Madison in a house designed by Frank University of Washington. His email address is stout@math. Lloyd Wright, and its intriguing architecture and washington.edu. two-story-high living room made it a center for DOI: http://dx.doi.org/10.1090/noti955 social life in the department. March 2013 Notices of the AMS 295 Walter Rudin inner functions was summarized in a series of NSF- was one of the CBMS lectures, which were then published in 1986 preeminent math- as New Constructions of Functions Holomorphic in ematicians of his the Unit Ball of Cn. generation. He Walter Rudin is also known to generations of worked in a num- undergraduate and graduate students for his three ber of different outstanding textbooks: Principles of Mathematical areas of mathe- Analysis (1953), Real and Complex Analysis (1966), matical analysis, and Functional Analysis (1973). In 1993 he was and he made ma- awarded the American Mathematical Society’s Photograph courtesy of MaryRudin. Ellen jor contributions Leroy P. Steele Prize for Mathematical Exposition. Rudin house, Madison. to each. His early He received an honorary degree from the University work reflected his classical training and focused on of Vienna in 2006. the study of trigonometric series and holomorphic In addition to his widow, Mary Ellen, Walter Rudin functions of one complex variable. He was also is survived by his four children: Catherine Rudin, very influenced by the then relatively new study of professor of modern languages and linguistics at Banach algebras and function algebras. One of his Wayne State College, Nebraska; Eleanor Rudin, an important results in this area, building on the work engineer working for 3M in St. Paul, Minnesota; of Arne Beurling, is the complete characterization Robert Rudin of Madison, Wisconsin; and Charles of the closed ideals in the disk algebra in 1956. Rudin, professor of oncology at the Johns Hopkins Another major area of Walter’s interest was University in Baltimore. He is also survived by four the general theory of harmonic analysis on locally grandchildren: Adem, Deniz, Sofia, and Natalie. compact Abelian groups. In the late 1950s and 1960s this was a very active and popular area Jean-Pierre Kahane of research, and perhaps only partially in jest, Walter suggested that mathematicians introduce a Walter Rudin and Harmonic Analysis new word, “lgbalcag”, to replace the phrase “Let The work of Walter Rudin on harmonic analysis is G be a locally compact Abelian group”, which a good part of his life and also of mine. A guide is how almost every analysis seminar began in for most of it is the list of his papers on Fourier those days. One of Walter’s major achievements in analysis on groups at the end of his celebrated this area was his 1959 work with Helson, Kahane, book. I shall start with a few of them and add and Katznelson, which characterized the functions some complements. This is nothing but a glance that operate on the Fourier transforms of the at a large piece of harmonic analysis. L1-algebra. Rudin synthesized this aspect of his The general inspiration for Walter was to dis- mathematical career in his 1962 book, Fourier cover questions and results arising from the Analysis on Groups. algebraic structure of some parts of analysis. Walter’s interests changed again in the late Wiener and Gelfand had paved the way. It became 1960s, and he began to work on problems in the main tendency in harmonic analysis in the several complex variables. At that time the study of middle of the twentieth century. the analytic aspects of complex analysis in several Part of Walter’s work deals with the Wiener variables was relatively new and unexplored, and algebras, that is, the algebras made of Fourier it was not even clear what the right several- transforms of integrable functions or summable variable generalization of the one-dimensional sequences. A related matter is trigonometric series. unit disk should be. There are at least two Another is convolution algebras of Radon measures. candidates: the polydisk and the ball. Walter A general framework is the Gelfand theory of did important work with both. For example, he Banach algebras. Generally speaking, there are showed for the polydisk (1967) and the unit ball several ways to express the same problems and (1976) that the zero sets of different Hp classes results: classical in the sense of the nineteenth of functions are all different. His work on the century or abstract and modern in the sense of “inner function conjecture” led to a tremendous the twentieth. Moreover, as far as Fourier analysis amount of research, and after the solution by is concerned, what happens on a group can be Aleksandrov and Hakim-Sibony-Løw (1981), Walter expressed on the dual group. Equivalent definitions made additional important contributions to this can be found everywhere, and Walter was an expert question. Much of Rudin’s work in several complex in playing this game. variables is presented in three of his advanced books. The first, published in 1969, is Function Jean-Pierre Kahane is emeritus professor of mathemat- Theory in Polydiscs. The second, published in 1980, ics at the Université de Paris-Sud. His email address is is Function Theory in the Unit Ball of Cn. His work on [email protected]. 296 Notices of the AMS Volume 60, Number 3 Let me start with the first paper of his mentioned to Lp(G∗) (here G∗ is the in Fourier Analysis on Groups. The title is “Non dual group of G), with p < analytic functions of absolutely convergent Fourier 2 (depending on f ), then F series”, and the year was 1955 [1]. What he dis- is the restriction on (−1; 1) covered was a positive function with an absolutely of an analytic function in a convergent Fourier series, vanishing at 0, such that neighborhood of the closed its square root does not enjoy the same property. interval [−1; 1]. It was the first result of this type, but the question The main event in the do- was in the air, if in different forms. The Wiener-Lévy main after 1958 was the theorem asserts that an analytic function of a func- theorem of Malliavin on tion in the Wiener algebra A(T) (that is, its Fourier spectral synthesis in 1959. series converges absolutely) belongs to A(T). In Spectral synthesis can be ex- other words, the analytic functions operate on pressed in many ways, as A(T); that is, the convolution algebra l1(Z) has a well as nonspectral synthesis. symbolic calculus consisting of analytic functions. The contribution of Walter in Can we replace the analytic functions by a wider that subject was to exhibit a function f in A(G) such that class? The Wiener-Lévy theorem can be translated Photograph courtesy of Mary Ellen Rudin.
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