Let $H$ be a separable Hilbert space. It is known that if $\{f_{n}\}$ is a frame then it is complete, but the converse is not true. In which cases the converse will be true, i.e.
When a complete sequence $\{f_{n}\}$ becomes a frame for $H$ (or at least satisfying the lower frame bound)?
Note:
(1) A sequence $\{f_{n}\}$ is a frame for a separable Hilbert space $H$ if there exists $A,B>0$ such that for all $f$ in $H$ A \lVert f\rVert^2 \leq \sum |\langle f,f_n\rangle|^{2}
(2) $\{f_{n}\}$ is complete if the only element which is orthogonal to all $f_{n}$ is the zero element.
Any comments or references are welcome!