LAUSR

Let f be the characteristic function of a probability measure μ f on ℝ n , and let σ > 0. We study the following extrapolation problem: under what conditions on the neighborhood of infinity V σ  = { x  ∈ ℝ n  : | x k | >  σ ,  k  = 1, … n } in ℝ n does there exist a characteristic function g on ℝ n , such that g = f on V σ but g ≢ f ? Let μ f have a nonzero absolutely continuous part with continuous density ???? . In this paper, we give certain sufficient conditions on ???? 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 ???? matter, but also certain arithmetic properties of supp ???? .