Rescaled localized radial basis functions and fast decaying polynomial reproduction

Approximating a set of data can be a difficult task but it is very useful in applications. Through a linear combination of basis functions we want to reconstruct an unknown quantity from partial information. We study radial basis functions (RBFs) to obtain an approximation method that is meshless, provides a data dependent approximation space and generalization to larger dimensions is not an obstacle. We analyze a rational approximation method with compactly supported radial basis functions (Rescaled localized radial basis function method). The method reproduces exactly the constants and the density of the interpolation nodes influences the support of the RBFs. There is a proof of the convergence in a quasi-uniform setting up to a conjecture: we can determine a lower bound for the approximant of the constant function 1 uniformly with respect to the size of the support of the kernel. We investigate the statement of the conjecture and bring some practical and theoretical results to support it. We study the Runge phenomenon on the approximant and obtain uniform estimates on the cardinal functions. We extend the distinguishing features of the method reproducing exactly larger polynomial spaces. We replace local polynomial reproduction with basis functions that decrease rapidly and approximate exactly a polynomial space. This change releases the basis functions from the compactness of the support and guarantees the same convergence rate (the oversampling problem does not appear). The rescaled localized radial basis function method can be interpreted in this new framework because the cardinal functions have global support even if the kernel has compact support. The decay of the basis functions undertake convergence and stability. In this analysis the smoothness of the approximant is not important, what matters is the "locality" provided by the fast decay. With a moving least squares approach we provide an example of a smooth quasi-interpolant. We continue trying to improve the performance of the method even when the weight functions do not have compact support. All the new theoretical results introduced in this work are also supported by numerical evidence.