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Let dμ ≥ 0 be a Borel measure on [0,∞) and An = ∞∫ 0 tndμ(t) < ∞ (n = 0, 1, 2, ...) be its moments. T. Carleman found sharp conditions on the magnitude of {An}∞0 for dμ to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if A′n = ∞∫ 0 tdμ1(t) are the moments of another measure, dμ1 ≥ 0, with lim sup n→∞ |An−A′n| 1 n = ρ < ∞, then the measure dμ − dμ1 is supported on the interval [0, ρ]. This result generalizes both the Carleman theorem and a theorem of J. Mikusiński. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem. §1. Preliminaries Definition 1.1. A sequence {An}0 of positive numbers is called logarithmically convex if An ≤ √ An−1An+1 (n = 1, 2, ...). Clearly, the moments An = ∞ ∫ 0 tdμ(t) of any Borel measure dμ ≥ 0, with An < ∞, form a logarithmically convex sequence. Definition 1.2. If {An}0 is an arbitrary sequence of positive numbers, then the convex regularization of {An}0 by means of logarithms, denoted {An}0 , is formed by setting An = sup{Bn : {Bν}0 is logarithmically convex, and Bν ≤ Aν (ν = 0, 1, ...)}. Proposition 1.3 ([Ma]). Given a sequence {An}0 , An > 0, the following three conditions are equivalent: Carleman’s condition: ∞ ∑ n=0 1 βn = ∞, where βn = inf k≥n A 1 k k ; (C) Received by the editors June 13, 1996 and, in revised form, December 10, 1996. 1991 Mathematics Subject Classification. Primary 30E05; Secondary 26E10.

A generalization of Carleman’s uniqueness theorem and a discrete Phragmén-Lindelöf theorem