Biographical Memoirs Volume 85 (2004) / Chapter Skim
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Salomon Bochner
Salomon Bochner
Pages 40-61
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1992. copyright Bochner, Salomon of Papers Collected Society.
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He contributed to the fields of almost periodic functions, classical Fourier analysis, complex analysis in one and several variables, differential geometry, Lie groups, probability, and history of science, among others. He did not often write long papers.
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The main theorem of Bohr's first Acta paper is that f is determined by its coefficients a. In the second Acta paper the main theorem is the desired result that any almost periodic function can be approximated uniformly by finite linear combinations of functions ei .
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Indeed, John von Neumann2 in 1934 published a generalization to all groups that used Bochner's definition and combined it with techniques from the work of Hermann Weyl. Bochner and von Neumann combined forces to write a sequel (1935)
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He paid particular attention to convergence questions and to the Poisson summation formula, which relates the Fourier transform to Fourier series and is used in proving the "modular relation" that connects the values of a theta function at z and 1/z in the upper half plane. His book Vorlesungen über Fouriersche Integrale (1932)
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Thus Bochner's Theorem becomes part of the foundation of the theory of locally compact abelian groups. The adele and idele groups of a number field furnish important examples of locally compact abelian groups, and these theorems for such groups are essential underpinnings in the modern understanding of class field theory.
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. A similar conclusion holds for Fourier transforms.
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For = 0, Stein obtained nothing new -- only the convergence in L2. Stein made critical use of an observation that although the restriction of the Fourier transform to a hyperplane does not make sense for an Lp function when p >1, there is a nontrivial interval 1 p < p0 such that restriction to a sphere makes sense for the Fourier transform of an Lp function.
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if a compact H with positive mean curvature is covered by a finite number of neighborhoods, if a meromorphic functional element is defined in each neighborhood and if the difference of meromorphic elements is holomorphic whenever the elements overlap, then there exists one meromorphic function on H which differs by a holomorphic function from each meromorphic element given. Bochner pursued this topic for five or six years, writing several papers, one of them joint with K
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In the late 1930s Bochner began a systematic investigation of functions of several complex variables. Robert Gunning, the editor of Bochner's collected papers and a student of Bochner's from the 1950s, summarizes this work as follows18: Bochner's interest in functions of several complex variables began with their Fourier analysis, leading to his characterization of the envelopes of holomorphy of tube domains [(1938)
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. was among the first on the subject of several complex variables; although there are now many books on the subject, that volume is frequently cited in the modern literature." About Bochner's work as a whole, Krantz continues, "Not only did Bochner touch many areas of mathematics, but his ideas are so profound that they are still of great interest today." Salomon Bochner, son of Joseph and Rude Bochner, was born on August 20, 1899, into a Jewish family of modest means in the Polish city of Krakow, which was then part of the Austro-Hungarian Empire.
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According to an online mathematics genealogy project,21 Bochner's thesis adviser was Erhard Schmidt. In later years Bochner would not say much about Schmidt.
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During this period in his life, Bochner was a part-time member of the Institute for Advanced Study for 1945-1948, a lecturer at Harvard for the spring semester of 1947, a consultant to the Los Alamos Project in Princeton in 1951, and for 1952-1953 a visiting professor in the Department of Statistics at the University of California, Berkeley. In 1959 Bochner was appointed Henry Burchard Fine Professor of Mathematics, and he held that position until his mandatory retirement from Princeton in 1968.
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He was an invited speaker at the International Congress of Mathematicians in 1950, gave the Colloquium Lectures of the American Mathematical Society in 1956, and was keynote speaker at the AAAS Symposium in 1971 on the "Role of Mathematics in the Development of Science." In January 1979 the American Mathematical Society awarded him the first Leroy P Steele Prize for Lifetime Achievement, citing him for "his cumulative influence on the fields of probability theory, Fourier analysis, several complex variables, and differential geometry." As Bochner grew older he partly turned from mathematics to classics, philosophy, and the history of science and of mathematics.
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Matthew Bochner Kennel is an assistant research physicist at the University of California, San Diego, and Sarah Alexandra Kennel is an assistant curator at the National Gallery of Art in Washington, D.C. Deborah described Salomon Bochner as a very attentive father, who gave lifelong unconditional love and, as she matured, intellectual stimulation and companionship in a wide variety of humanistic subjects.
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Bochner pushed Langlands in the direction of algebraic number theory, arranging for him to teach a course in class field theory. One of Bochner's thesis students, William Veech, remembers passing Langlands in the hall one day in the 1960s and asking Langlands what he would do next.
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The advice he offered was more philosophical, or sometimes sociological, than mathematical. Mathematical advice was left to another, earlier Bochner student, Harry Furstenberg, who was visiting Princeton for a year.
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I am indebted to Deborah Bochner Kennel for extensive help in preparing this article and to Robert Gunning and William Veech for offering useful information and comments.
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Collected Papers of Salomon Bochner. Parts 14.
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1933 Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind.
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52:776-797. Linear partial differential equations with constant coefficients.
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72:3815 3818. 1979 Fourier series came first.
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