Successive approximation by spaces of local functions

One considers interpolation and approximation by local functions on a nonuniform net and stable algorithms for their successive construction. One introduces the concept of $H$-stability of a family of interpolations and one determines sufficient conditions for $H$-stability. One constructs nonhomogeneous spaces of local functions which realize an a priori approximation of a given order; the basis functions of these spaces have in a certain sense a minimal support and possess interpolation properties.