American mathematician Salomon Bochner

From McGraw-Hill encyclopedia of science and technology. McGraw-Hill Modern Men of Science; 426 Leading Contemporary Scientists Presented by the Editors of the McGraw-Hill Encyclopedia of Science and Technology. McGraw-Hill, 1966.

BOCHNER, Salomon

American mathematician

Born August 20, 1899, Cracow, Poland (then Austria-Hungary)

Died 2 May 1982 Houston, Texas  

Bochner's achievements were mainly in the field of harmonic or Fourier analysis, that is, in the theory of representation of general functions by trigonometric series, trigonometric integrals, and more comprehensive expansions. But he was also active in the theory of functions of several complex variables, especially on Cauchy formulas, and on functions in tubular domains, which latter topic has applications to hyperbolic differential equations and to problems in quantum field theory. In differential geometry he introduced a general topic that goes under the name "curvature and Betti numbers."

In physics and technology many phenomena are viewed as "waves" (which subsumes "vibrations" or "oscillations"); there are waves in acoustics, hydronamics, electrodynamics, optics, and quantum mechanics, among others. Invariably, a "general" wave is an additive, finite or infinite, superposition of "simple" waves, a simple wave being one with a specific single "frequency" is associable; also invariably, in each context there is a certain "typical" case, in which all occurring frequencies are integer-values multiples of a single one, which is then the smallest one. This skeletal "sameness" within physical contexts of different provenance, especially the sameness of the "typical" case, is brought about by the fact that, in each case, a certain function f(x), the wave function (say on an interval of length 2π), is represented as a series, with constant coefficients, of the "simple" functions { sin nx, cos nx } or, in the complex version, of the "simple" functions  { e^imx }. This mathematical expansion is of course independent of the physical interpretation in which the function f(x) may be involved.

Another "typical" case arises if, as J.B. Fourier did in the early 19th century, we vary the interval over which f(x) is defined, and then let the length of the interval tend to infinity. This replaces the trigonometric series by a trigonometric integral, that is, the Fourier coefficients by a certain function, say 𝝋(ɑ), which is called the Fourier transform of f(x). Now the relation between between f(x) and its transform 𝝋(ɑ) is a "dual" or "reciprocal" one, meaning that f(x) can be reobtained from 𝝋(ɑ) in more or less the same manner as 𝝋(ɑ) was obtained from f(x). After lying dormant for the better part of a century, this duality began to be studied in earnest at the beginning of the 20th century. The principle underlying it is so general and comprehensive that the awareness and presence of it was felt in ever wider and ever more numerous areas of mathematics and physics. For instance, if formulated in a suitably comprehensive version, this duality subsumes the de Broglie duality between waves and corpuscles.

A new event occurred in the early 1920s when the mathematician Harold Bohr (younger brother of the physicist Niels Bohr) introduced, on -∞ < x < ∞, so-called almost periodic functions (which had been foreshadowed as far back as in the mechanics of Lagrange), each of which has a "discrete" Fourier expansion, in which however the "frequencies" λ , which occur in the "simple" functions { e^iλx }, can be arbitrary real numbers, with no requirement at all of commensurateness between any two of them.

Bochner soon introduced an algorithmic summability process, the so-called Bochner-Fejer process, which makes it just as easy to handle the Fourier expansion of almost periodic functions-for any type of almost periodicity, however general-as the customary case of periodic functions. Furthermore, Bochner gave an entirely different characterization of the class of almost periodic function on the line, namely, by a certain topological property of compactness. It was this alternate definition that afterwards enabled John von Neumann to extend almost periodicity from the euclidean line to other group spaces.

In the field of Fourier integrals, Bochner found a much applied criterion for a continuous complex-valued function 𝝋(ɑ) to be representable as Fourier-Stieltjes integral 

𝝋(ɑ) = ∫ e^iaxdF(x)

in which dF ≥ 0. For this to be possible it is not only  necessary but also sufficient that 𝝋(ɑ) be positive-definite, meaning that for any finitely many points ɑ₁,…, ɑ_n , and complex constants  c₁,…, c_n , there is:

𝚺   (p,q=1…n) c_pc_q𝝋 (ɑ_p-ɑ_q ) ≥ 0

This criterion has several applications in the theory of probability; it is also usable for the derivation of the spectral representation of a self-adjoint operator in Hilbert space; and it has been generalized and applied to functions on topological group spaces.

Bochner was also a precursor in the theory of the so-called Schwartz distributions in that he introduced generalized Fourier transforms for functions that do not grow faster at infinity than a power of x.

After attending high school in Berlin, Bochner studied mathematics at the University of Berlin, terminating with a Ph.D. in 1921. In 1924-26 he studied, partly as a fellow of the International Education Board, with Harold Bohr in Copenhagen, and with G. H. Hardy and J. E. Littlewood in Oxford and Cambridge. In 1926-33 he was lecturer at the University of Munich, where he wrote his first book on Fourier integrals. In 1933 he joined the mathematics department of Princeton University and in 1951 became Henry Burchardt Fine Professor there. He was elected to the National Academy of Sciences in 1950.

Bochner wrote "Several Complex Variables" (1948), "Fourier Transforms" (1949), "Curvature and Betti Numbers" (1953), "Harmonic Analysis and the Theory of Probability" (1955), "Fourier Integrals" (1959), and "The Role of Mathematics in the Rise of Science" (1966).

For background information see "Analysis of Variance"; "Fourier Series and Integrals"; "Integral Transform" in the McGraw-Hill Encyclopedia of Science and Technology.