Riesz-Fischer Sequences and Lower Frame Bounds

Abstract

We investigate the consequences of the lower frame condition and the lower Riesz basis condition without assuming the existence of the corresponding upper bounds. We prove that the lower frame bound is equivalent to an expansion property on a subspace of the underlying Hilbert space $\mathcal{H}$, and that the lower frame condition alone is not enough to obtain series representations on all of $\mathcal{H}$. We prove that the lower Riesz basis condition for a complete sequence implies the lower frame condition and $\omega$-independence; under an extra condition the statements are equivalent.