Harmonic Analysis and the Theory of Probability

Harmonic Analysis and the Theory of Probability by Salomon Bochner is a landmark monograph that rigorously links two central domains of modern mathematics: Fourier analysis and probability theory. Written during the author’s time at the Statistical Laboratory at Berkeley under the direction of Jerzy Neyman, the book develops a unified account of approximation, Fourier expansions, Laplace and Mellin transforms, stochastic processes, and their deep interrelations. Bochner’s treatment moves fluidly from classical tools—kernels, summability formulas, spherical harmonics—to more advanced concepts such as infinitely divisible processes, characteristic functionals, and the closure properties of Fourier transforms, always with an eye to their probabilistic interpretations.

This volume demonstrates how probability theory gains analytic depth when recast in the language of harmonic analysis, and how harmonic methods are enriched by probabilistic intuition. It offers precise theorems on convergence, moment conditions, and stochastic functionals, while also exploring zeta integrals, random paths, and the analysis of stationary processes. As part of the California Monographs in Mathematical Sciences series, it speaks to mathematicians, statisticians, and theoretical physicists seeking rigorous frameworks for randomness and structure. Bochner’s synthesis not only advanced mid-twentieth-century probability but also laid groundwork for contemporary research in functional analysis, statistical mechanics, and stochastic modeling.

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1955.