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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 couse, 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!

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Frames and completeness

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!