The Approximate Inverse in Action IV: Semi-Discrete Equations in a Banach Space Setting

This article concernes the method of approximate inverse to solve semi-discrete, linear operator equations in Banach spaces. Semi-discrete means that we search a solution in an infinite dimensional Banach space having only a finite number of data available. In this sense the situation is applicalble to a large variety of applications where a measurement process delivers a discretization of an infinte dimensional data space. The method of approximate inverse computes scalar products of the data with pre-computed reconstruction kernels which are associated with mollifiers and the dual of the model operator. The convergence, approximation power and regularization property of this method when applied to semi-discrete operator equations in Hilbert spaces has been investigated in three prequels of that article. Here we extend these results to a Banach space setting. We show convergence and stability and reproduce the results for the integration operator acting on the space of continuous functions.