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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.
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 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.
Let (,,) be a measure space, and be a Banach space.The Bochner integral of a function : is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form = = (), where the are disjoint members of the -algebra , the are distinct elements of , and χ E is the characteristic function of .
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (,) to the Ricci ... (by the divergence theorem) ...
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.
Alexandrov theorem; Almgren regularity theorem; Area formula (geometric measure theory) B. ... Bochner's theorem; Borel–Cantelli lemma; Brunn–Minkowski theorem; C.