Redundant Representation of Operators

To be able to solve operator equations numerically a discretization of those operators is necessary. In the Galerkin approach bases are used to achieve discretized versions of operators. In a more general set-up, frames can be used to sample the involved signal spaces and therefore those operators. Here we look at the redundant representation of operators resulting from a matrix representation using frames. We focus on injectivity, surjectivity and, in particular, invertibility of the involved operators and matrices. Furthermore we show sufficient conditions that the composition of matrices correspond to the composition of operators.