Abstract
We study an extension property for characteristic functions f : Rn → C of probability measures. More precisely, let f be the characteristic function of a probability density φ on Rn, and let Uσ = {x ∈ Rn: mink|xk| > σ}, σ > 0, be a neighborhood of infinity. We say that f has the σ-deterministic property if for any other characteristic function g such that f = g on Uσ, it follows that f ≡ g. A sufficient condition on f to has the σ-deterministic property is given. We also discuss the question about how precise our sufficient condition is? These results show that the σ-deterministic property of f depends on an arithmetic structure of the support of φ.
Highlights
Let M (Rn) be the family of finite complex-valued regular Borel measures on Rn
For a characteristic function f and a subset U of Rn, we study the problem: is it true that there exists a characteristic function g on Rn such that g = f on U but g ≡ f ? Our interest to this question is initiated by a similar problem posed by N.G
Let f : Rn → C be the characteristic function of a probability density φ
Summary
Let M (Rn) be the family of finite complex-valued regular Borel measures on Rn. Given a measure μ ∈ M (Rn), we define its Fourier transform by μ(x) = e−i(x,t) dμ(t), Rn x ∈ Rn. Let f : R → C be the characteristic function of a distribution with a continuous and strictly positive density. There exists, for each σ > 0, a characteristic function g such that f (x) = g(x) if x = 0 or |x| σ and f (x) = g(x) otherwise.
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