Stabilization of spline bases by extension

We present a method to stabilize bases with local supports by means of extension. It generalizes the known approach for tensor product B-splines to a much broader class of functions, which includes hierarchical and weighted variants of polynomial, trigonometric, and exponential splines, but also box splines, T-splines, and other function spaces of interest with a local basis. Extension removes elements that cause instabilities from a given basis by linking them with the remaining ones by means of a specific linear combination. The two guiding principles for this process are locality and persistence. Locality aims at coupling basis functions whose supports are close together, while persistence guarantees that a given set of globally supported functions, like certain monomials in the case of polynomial splines, remain in the span of the basis after extension. Furthermore, we study how extension influences the approximation power and the condition of Gramian matrices associated with the basis, and present a series of examples illustrating the potential of the method.