On an extrapolation problem for characteristic functions

Abstract

Let f be the characteristic function of a probability measure μ_f on ℝⁿ, and let σ > 0. We study the following extrapolation problem: under what conditions on the neighborhood of infinity V_σ = {x ∈ ℝⁿ : |x_k| > σ, k = 1, …n} in ℝⁿdoes there exist a characteristic function g on ℝⁿ, such that g = f on V_σ but g ≢ f? Let μ_f have a nonzero absolutely continuous part with continuous density 𝜑. In this paper, we give certain sufficient conditions on 𝜑 and V_σ under which the latter question has an affirmative answer. We also address the optimality of these conditions. Our results indicate that not only does the size of both V_σ and the support supp 𝜑 matter, but also certain arithmetic properties of supp 𝜑.