Generating sets for Beurling algebras

We characterize in terms of Beurling–Malliavin density, the generating sets for Beurling algebras , that is the sets Λ⊂R for which a function exists such that the Λ-translates {ϕ(x-λ)},λ∈Λ, span . Our main result extends a recent theorem from [J. Bruna, A. Olevskii, A. Ulanovskii, Completeness in L1(R) of discrete translates, arXiv:math.CA/0307323v1, 2003, (Revista Mathematica Iberoamericana), submitted for publication.], which describes the generating sets for L1(R).