Cesare Bracco

Dimensions and bases of hierarchical tensor-product splines

We prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a three-dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a certain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space. This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Juttler for two-dimensional hierarchical meshes.

Multivariate Hermite–Birkhoff interpolation by a class of cardinal basis functions

Abstract A class of cardinal basis functions for Hermite–Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard’s functions is finally discussed, including some numerical examples.

Spaces of generalized splines over T-meshes

We consider a class of non-polynomial spaces, namely a noteworthy case of Extended Chebyshev spaces, and we generalize the concept of polynomial spline space over T-mesh to this non-polynomial setting: in other words, we focus on a class of spaces spanned, in each cell of the T-mesh, both by polynomial and by suitably-chosen non-polynomial functions, which we will refer to as generalized splines over T-meshes. For such spaces, we provide, under certain conditions on the regularity of the space, a study of the dimension and of the basis, based on the notion of minimal determining set, as well as some results about the dimension of refined and merged T-meshes. Finally, we study the approximation power of the just constructed spline spaces.

Generalized T-splines and VMCR T-meshes

Abstract The paper considers the extension of the T-spline approach to the Generalized B-splines (GB-splines), a relevant class of non-polynomial splines. The Generalized T-splines (GT-splines) are based both on the framework of classical polynomial T-splines and on the Trigonometric GT-splines (TGT-splines), a particular case of GT-splines. Our study of GT-splines introduces a class of T-meshes (named VMCR T-meshes) for which both the corresponding GT-splines and the corresponding polynomial T-splines are linearly independent. A practical characterization can be given for a sub-class of VMCR T-meshes, which we refer to as weakly dual-compatible T-meshes, which properly includes the class of dual-compatible (equivalently, analysis-suitable) T-meshes for an arbitrary (polynomial) order.

Bases of T-meshes and the refinement of hierarchical B-splines

Abstract In this paper we consider spaces of bivariate splines of bi-degree ( m , n ) with maximal order of smoothness over domains associated to a two-dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor-product B-splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Juttler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T-mesh the hierarchical B-splines form a basis of bivariate splines of bi-degree ( m , n ) with maximal order of smoothness over this hierarchical T-mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.

Lagrange Interpolation on Arbitrarily Distributed Data in Banach Spaces

The Lagrange interpolation problem in Banach spaces is approached by cardinal basis interpolation. Some error estimates are given and the results of several numerical tests are reported in order to show the approximation performances of the proposed interpolants. A comparison between some examples of interpolants is presented in the noteworthy case of Hilbert spaces, with some considerations about the possible localization of the formulas. Finally, some remarks about the cardinal basis interpolation framework are made from the application point of view.

Iterative refinement of hierarchical T-meshes for bases of spline spaces with highest order smoothness

In this paper we propose a strategy for generating consistent hierarchical T-meshes which allow local refinement and offer a way to obtain spline basis functions with highest order smoothness incrementally. We describe the required ordering of line-segments during refinement and the construction of spline basis functions. We give our strategy for generating consistent hierarchical T-meshes over any shape of a two-dimensional domain.

Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data in Banach Spaces

A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Fréchet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation.

Tchebycheffian spline spaces over planar T-meshes: Dimension bounds and dimension instabilities

Abstract We consider Tchebycheffian spline spaces over planar T-meshes and we study their dimension. We show that the structure of extended Tchebycheff spaces allows us to fully generalize the dimension upper bounds known in the literature for polynomial spline spaces over T-meshes. Moreover, we illustrate that the dimension of Tchebycheffian spline spaces over T-meshes can be unstable for certain configurations of the T-mesh, for any choice of the underlying extended Tchebycheff space.