On the completeness of hierarchical tensor-product B-splines

Given a grid in R, consisting of d bi-infinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensor-product spline functions of a given degree on a multi-cell domain. Such a domain consists of finite set of cells which are defined by the grid. A piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid. We prove that the connected components of the associated set of tensor-product B-splines, whose support intersects the multi-cell domain, form a basis of this spline space. More precisely, if the intersection of the support of a tensor-product B-spline with the multi-cell domain consists of several connected components, then each of these components contributes one basis function. In order to establish the connection to earlier results, we also present further details relating to the three-dimensional case with single knots only. A hierarchical B-spline basis is defined by specifying nested hierarchies of spline spaces and multi-cell domains. We adapt the techniques from [12] to the more general setting and prove the completeness of this basis (in the sense that its span contains all piecewise polynomial functions on the hierarchical grid with the smoothness specified by the grid and the degrees) under certain assumptions on the domain hierarchy. Finally, we introduce a decoupled version of the hierarchical spline basis that allows to relax the assumptions on the domain hierarchy. In certain situations, such as quadratic tensor-product splines, the decoupled basis provides the completeness property for any choice of the domain hierarchy.