Bessel sequences in Sobolev spaces

In this paper we investigate Bessel sequences in the space L2(R s), in Sobolev spaces Hμ(Rs) (μ > 0), and in Besov spaces B μ p,p(R s) (1 p ∞). For each j ∈ Z, let Ij be a countable index set. Let (ψj,α)j∈Z, α∈Ij be a family of functions in L2(R). We give some sufficient conditions for the family to be a Bessel sequence in L2(R s) or Hμ(Rs). The results obtained in this paper are useful for the study of frames and Riesz bases for L2(R s) or Hμ(Rs). In particular, these results are applicable to wavelets on irregular meshes. © 2005 Elsevier Inc. All rights reserved.