BochnerTheorem

Bochner's theorem, named after the mathematician Salomon Bochner, is a fundamental result in harmonic analysis and probability theory. It describes when a function can be interpreted as the Fourier transform of a measure, linking positive-definite functions to finite measures on dual groups and revealing a deep connection between analysis and probability.

In the setting of locally compact abelian groups, let G be such a group and let Ĝ denote its dual group of continuous unitary characters. A function φ: G → C is continuous, positive-definite, and satisfies φ(0) = 1 if and only if there exists a unique finite positive Borel measure μ on Ĝ such that φ(g) = ∫ Ĝ χ(g) dμ(χ) for every g in G. Here φ is the Fourier-Stieltjes transform of μ. When G is R^n with its standard dual, this specializes to the classical statement about characteristic functions.

In the Euclidean setting, φ is the characteristic function of a probability distribution on R^n precisely when φ is positive-definite, continuous at 0, and φ(0) = 1. Equivalently, φ(t) = ∫ e^{i t·x} dμ(x) for a unique probability measure μ. The theorem thus characterizes which functions arise as Fourier transforms of finite measures and, in probability terms, which functions are distributional characteristic functions.

Bochner's theorem has wide applications in probability, statistics, and harmonic analysis, including the study of stochastic processes, spectral representations, and the construction of probability measures from known characteristic functions. It remains a foundational result illustrating the deep link between positive-definite functions and measures on dual groups. See also positive-definite functions, Fourier transform, and characteristic function.