Generalized frame operator, lower semiframes, and sequences of translates

Given an arbitrary sequence of elements ξ = { n } ∈ N $\xi =\lbrace \xi _n\rbrace _{n\in \mathbb {N}}$ a Hilbert space ( H , ⟨ · ⟩ ) $(\mathcal {H},\langle \cdot ,\cdot \rangle )$ the operator T $T_\xi$ is defined as associated to sesquilinear form Ω f g ∑ $\Omega _\xi (f,g)=\sum {N}} \langle _n\rangle _n g\rangle$ for h : | 2 < ∞ $f,g\in \lbrace h\in \mathcal {H}: \sum {N}}|\langle |^2<\infty \rbrace$ . This in general different from classical frame but possesses some remarkable properties. For instance, always self-adjoint with regard particular space, unconditionally defined, and, when lower semiframe, gives simple expression dual ξ. The and semiframes are studied context sequences integer translates function L R $L^2(\mathbb {R})$ In particular, explicit given this context, characterization translates, which semiframes, proved.