Related papers: An algebraic framework for geometrically continuou…
Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline…
Periodic splines are a special kind of splines that are defined over a set of knots over a circle and are adequate for solving interpolation problems related to closed curves. This paper presents a method of implementing the objects…
We construct over a given bilinear multi-patch domain a novel $C^s$-smooth mixed degree and regularity isogeometric spline space, which possesses the degree $p=2s+1$ and regularity $r=s$ in a small neighborhood around the edges and…
This paper introduces a novel framework for constructing $C^r$ basis functions for polynomial spline spaces of degree $d$ over arbitrary planar polygonal partitions, overturning the belief that basis functions cannot be constructed on…
A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial…
This paper develops a new way to create smooth piecewise polynomial free-form spline surfaces from quad- meshes that include T-junctions, where surface strips start or terminate. All mesh nodes can be interpreted as control points of…
A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of…
Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometrically-oriented compounds. Box splines are an…
Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate…
We consider overlap splines that are defined by connecting the patches of piecewise functions via common values at given finite sets of nodes, without using any partitions of the computational domain. It is shown that some classical finite…
The frame set of a function $g\in L^2(\mathbb{R})$ is the subset of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ In this…
Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If…
One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from $p$-degree splines (and extensions, such as NURBS), they enjoy up to $C^{p-1}$ continuity within each…
We introduce here a direct method to construct multivariate explicit B-spline bases. B-splines are piecewise polynomials, which are defined on adjacent tetrahedra and which are $C^{r}$ continuous throughout. The $C^{r}$ continuity is…
Graphical (Linear) Algebra is a family of diagrammatic languages allowing to reason about different kinds of subsets of vector spaces compositionally. It has been used to model various application domains, from signal-flow graphs to Petri…
This paper discusses the dimension of spline spaces with highest order smoothness over hierarchical T-meshes over certain type of hierarchical T-meshes. The major step is to set up a bijection between the spline space with highest order…
We consider a class of non-polynomial spline spaces over T-meshes, that is, of spaces locally spanned both by polynomial and by suitably-chosen non-polynomial functions, which we will refer to as generalized splines over T-meshes. For such…
Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process.…
Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\Delta$, for $d \geq 3r+1$.…
Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial…