Frame bounds for Subspaces of L2

Let H be a separable Hilbert space with inner product (•,•). We say a set {f _k: k ∈ ℤ} is a (fundamental) frame for H if there exist A,B > 0 such that for each f ∈ H, A ∥ f ∥ ² _H ≤ Σ _n∈ℤ|(f,f _k)| ²≤B∥f∥ ² _H In case {f _k: k ∈ ℤ} is a frame for the subspace span{f _k: k ∈ ℤ}, we say that {f _k: k ∈ ℤ} is a frame sequence. A Weyl-Heisenberg frame sequence is a frame sequence which is generated by translated and modulated versions of L ₂-functions. In this paper, we characterize Weyl-Heisenberg frame sequences using infinite Hermitian matrices and obtain the optimal frame bounds in terms of the operator norms of these matrices. This work is inspired by a paper by Casazza and Christensen, where sufficient conditions for a Weyl-Heisenberg system to be a frame sequence are studied.