The linear algebra of interpolation with finite applications giving computational methods for multivariate polynomials

Abstract

Linear representation and the duality of the biorthonormality relationship express the linear algebra of interpolation by way of the evaluation mapping. In the finite case the standard bases relate the maps to Gramian matrices. Five equivalent conditions on these objects are found which characterize the solution of the interpolation problem. This algebra succinctly describes the solution space of ordinary linear initial value problems. Multivariate polynomial spaces and multidimensional node sets are described by multi-index sets. Geometric considerations of normalization and dimensionality lead to cardinal bases for Lagrange interpolation on regular node sets. More general Hermite functional sets can also be solved by generalized Newton methods using geometry and multi-indices. Extended to countably infinite spaces, the method calls upon theorems of modern analysis.