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
In the Euclidean setting, φ is the characteristic function of a probability distribution on R^n precisely when
Bochner's theorem has wide applications in probability, statistics, and harmonic analysis, including the study of stochastic