The Beurling-malliavin Density of a Random Sequence

A formula is given for the completeness radius of a random exponential system {tleiξnt}pn−1 l=0,n∈Z in terms of the probability measures of ξn. The purpose of this note is to formulate a probabilistic version of the celebrated Beurling-Malliavin theorem on the completeness radius of a sequence of complex exponentials [1]. First, to state the Beurling-Malliavin theorem, let Λ = {λn}n∈Z be a sequence of real numbers and {pn}n an associated sequence of positive integers, so that pn denotes the multiplicity of λn in Λ. The completeness radius R(Λ) of E(Λ) = {tleiλnt}pn−1 l=0,n∈Z is defined as the supremum over all R ≥ 0 such that E(Λ) is a complete sequence in L2(−R,R). The Beurling-Malliavin theorem states that R(Λ) = πD(Λ), where D(Λ) is the Beurling-Malliavin density of Λ. Following [3], we define this density as follows. A system of intervals {(an, bn)}n∈N, an < bn, is substantial if either 0 < a1 < b1 ≤ a2 < b2... and ∑ n∈N a −2 n (bn − an) = ∞, or 0 > b1 > a1 ≥ b2 > a2... and ∑ n∈N |bn|−2(bn − an) = ∞. If Λ has a finite accumulation point, then D(Λ) = ∞. Otherwise, D(Λ) is defined as the supremum over allR ≥ 0 for which there exists a substantial sequence of intervals {(an, bn)}n∈N such that the number of elements of Λ on each interval (an, bn) is greater than or equal to R(bn − an). If Ξ = {ξn}n∈Z is a sequence of independent random variables, we may study completeness problems for the random exponential system {tl exp(iξnt)}n l=0,n∈Z, where, as above, pn < ∞ denotes the multiplicity of the point ξn. Recently, this approach has proved to be fruitful and has led to new insight into classical problems [2], [4]. While [2], [4] deal with random perturbations of a fixed sequence, we now address the problem of computing the Beurling-Malliavin density of an arbitrary random sequence. To solve this problem, we need the Beurling-Malliavin density of a Borel (not necessarily locally finite) measure. If μ is not locally finite, we set d(μ) = ∞. Otherwise, the Beurling-Malliavin density d(μ) is defined as the supremum over all R ≥ 0 for which there exists a substantial sequence {(an, bn)}n∈N such that μ((an, bn)) ≥ R(bn − an) for every n ∈ N. Observe that to any sequence Λ = {λn}n∈Z one can associate a measure ν = ∑ n∈Z δλn , where δx is a unit measure concentrated at Received by the editors September 8, 1995 and, in revised form, December 11, 1995. 1991 Mathematics Subject Classification. Primary 42A61, 42C30. Research supported by NATO linkage grant LG 930329. c ©1997 American Mathematical Society