Free multivariate splines (Q1092247)

An interesting type of nonlinear approximation is the univariate approximation by splines of a given degree, with a certain number of free knots. The word free here means that the knots are allowed to vary with the function being approximated; only the number of knots is held fixed. There has been no useful formulation of free knot splines in the case of several variables. The authors introduce a definition of multivariate free spline approximation, generalizing the univariate notion of spline with free knots. They give a characterization of certain approximation classes of functions, which have a prescribed order of approximation in \(L_ p\), in terms of the classical Besov spaces, in a way that is similar to the univariate case.