Lagrange interpolation on arbitrarily distributed data in Banach spaces

The Lagrange interpolation problem is a classical topic of numerical analysis and has been widely developed in the real space setting. However, it has never been efficiently, constructively extended to Banach spaces. The goal of this thesis is to literally extend Shepard's method to Banach spaces and, throughout numerical tests, evaluate its efficiency and accuracy. First, we consider Prenter's approach and discover that it is not so applicable but more theoretical. Therefore, the generalization of the Shepard-type interpolant in Banach spaces appears more practical and gives good approximation results. We also extend our problem in Hilbert spaces, as particular cases of Banach spaces, in which we consider the norm induced by an inner product. In Hilbert spaces Prenter gives a more useful method in terms of inner product but it does not give satisfiable results, unless we use the localized version in which only the two closest nodes to the evaluation point are considered in the implementation of the interpolant. The Shepard-type interpolant yields to better approximation results which also improve if the localized version has been calculated. Here we show a very interesting example that approximate the Volterra operator.