On an extrapolation problem for characteristic functions
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Lithuanian Mathematical Journal
On an extrapolation problem for characteristic functions Saulius Norvidas Institute of Data Science and Digital Technologies, Vilnius University, Akademijos str. 4, Vilnius LT-04812, Lithuania (e-mail: [email protected]) Received March 1, 2019; revised April 14, 2020
Abstract. Let f be the characteristic function of a probability measure μf on Rn , and let σ > 0. We study the following extrapolation problem: under what conditions on the neighborhood of infinity Vσ = {x ∈ Rn : |xk | > σ, k = 1, . . . , n} in Rn does there exist a characteristic function g on Rn such that g = f on Vσ but g ≡ f ? Let μf have a nonzero absolutely continuous part with continuous density ϕ. In this paper, we give certain sufficient conditions on ϕ and Vσ 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 Vσ and the support supp ϕ matter, but also certain arithmetic properties of supp ϕ. MSC: 60E10, 32D15, 42A38 Keywords: characteristic function, density function, entire function, probability measure, Bernstein space
1 Introduction A function f : Rn → C is said to be positive definite if nj,k=1 f (xj − xk )cj ck 0 for all finite sets of complex numbers c1 , . . . , cn and points x1 , . . . , xn ∈ Rn . Any such a function f satisfies f (−x) = f (x) and |f (x)| f (0) for all x ∈ Rn . The Bochner theorem gives a description of continuous positive definite functions in terms of the Fourier transform. For this reason, let us recall certain notions. Let M (Rn ) be the Banach algebra of bounded regular complex-valued Borel measures μ on Rn . As usual, M (Rn ) is equipped with the total-variation norm μ. For μ ∈ M (Rn ), we define the Fourier transform by μ ˆ(x) = e−i(x,t) dμ(t), x ∈ Rn , Rn
where (x, t) = nk=1 xk tk is the scalar product on Rn . We identify L1 (Rn ) with the closed ideal in M (Rn ) of all measures that are absolutely continuous with respect to the Lebesgue measure dt = dt1 · · · dtn on Rn . Let μ ∈ M (Rn ) be a positive measure. If μ = 1, then in the language of probability theory, μ and f (x) := μ ˆ(x), x ∈ Rn , are called a probability measure and its characteristic function, respectively. In particular, if μ = ϕ dt with ϕ ∈ L1 (Rn ) such that ϕL1 (Rn ) = 1 and ϕ 0 on Rn , then ϕ is called the probability density function of μ or the probability density for short. The Bochner theorem (see, e.g., [7, p. 57]) states that a function f : Rn → C is the characteristic function of a probability measure if and only if f is continuous and positive definite on Rn with f (0) = 1. c 2020 Springer Science+Business Media, LLC 0363-1672/20/6003-0001
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S. Norvidas
Given a characteristic function f , we are interested in the question: Is it true that for any open subset U of Rn , 0 ∈ U , there exists a characteristic function g such that g = f on U but g ≡ f ? Our interest to this question is initiated by a similar problem posed by Ushakov [7, p. 276]: Is it2 tru
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