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In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group.
The theorem is a corollary of Bochner's more fundamental result which says that on any connected Riemannian manifold of negative Ricci curvature, the length of a nonzero Killing vector field cannot have a local maximum. In particular, on a closed Riemannian manifold of negative Ricci curvature, every Killing vector field is identically zero.
An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of ″nice″ Banach spaces.
Bochner’s theorem. An arbitrary function φ : R n → C is the characteristic function of some random variable if and only if φ is positive definite , continuous at the origin, and if φ (0) = 1 .
Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. See Kriging.
Bochner's theorem on Fourier transforms appeared in a 1932 book. His techniques came into their own as Pontryagin duality and then the representation theory of locally compact groups developed in the following years.
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (,) to the Ricci ... (by the divergence theorem) ...
Carathéodory kernel theorem; Carathéodory's theorem (conformal mapping) Carleson–Jacobs theorem; Carlson's theorem; Casorati–Weierstrass theorem; Cauchy–Hadamard theorem; Cauchy's integral formula; Cauchy's integral theorem; Classification of Fatou components; Complex conjugate root theorem; Corona theorem