Overcomplete Reproducing Pairs

The Gaussian Gabor system at the critical density has the property that it is overcomplete in $L^2(\mathbf{R})$ by exactly one element, and if any single element is removed then the resulting system is complete but is not a Schauder basis. This paper characterizes systems that are overcomplete by finitely many elements. Among other results, it is shown that if such a system has a reproducing partner, then it contains a Schauder basis. While a Schauder basis provides a strong reproducing property for elements of a space, the existence of a reproducing partner only requires a weak type of representation of elements. Thus for these systems weak representations imply strong representations. The results are applied to systems of weighted exponentials and to Gabor systems at the critical density. In particular, it is shown that the Gaussian Gabor system does not possess a reproducing partner.