Let $H$ be a separable Hilbert space.

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 $H$ $$ A\|f\|^2 \leq \sum |\langle f, f_n \rangle|^2 \leq B \|f\|^2 $$ $\{f_{n}\}$ is complete if the only element which is orthogonal to all $f_{n}$ is the zero element.

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)?

Of couse, converse is true if $f_n$ is orthonormal. But orthogonality is a very strong condition for me.

Any comments or references are welcome!

Let $H$ be a separable Hilbert space.

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 $H$ $$ A\|f\|^2 \leq \sum |\langle f, f_n \rangle|^2 \leq B \|f\|^2 $$ $\{f_{n}\}$ is complete if the only element which is orthogonal to all $f_{n}$ is the zero element.

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 does a complete sequence $\{f_{n}\}$ becomes a frame for $H$ (or at least satisfying the lower frame bound)?

Of course the converse is true if $f_n$ is orthonormal. But orthogonality is a very strong condition for me.

Any comments or references are welcome!