Dimensions of spline spaces over non-rectangular T-meshes

Spline spaces over rectangular T-meshes have been discussed in many papers. In this paper, we consider spline spaces over non-rectangular T-meshes. The dimension formulae of spline spaces over special simply connected T-meshes have been obtained. For T-meshes with holes, we discover a new type of dimension instability. We construct a relationship between the dimension of the spline space over a T-mesh \(\mathcal {T}\) with holes and the dimension of the spline space over a simply connected T-mesh associated with \(\mathcal {T}\).     

On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C ^d?1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula.     

Spline spaces on TR-meshes with hanging vertices

Polynomial spline spaces defined on mixed meshes consisting of triangles and rectangles are studied for the C ⁰ case. These include triangulations with hanging vertices as well as T-meshes. In addition to dimension formulae, explicit basis functions are constructed, and their supports and stability are discussed. The approximation power of the spaces is also treated.     

三维Ⅱ型剖分上的样条空间

三维Ⅱ型剖分上的样条空间施锡泉（大连理工大学数学研究所）ＳＰＬＩＮＥＳＰＡＣＥＳＯＮＴＹＰＥ－２ＴＲＩＡＮＧＵＬＡＴＩＯＮＩＮＲ￣３￥ＳｈｉＸｉ－ｑｕａｎ（Ｉｎｓｔ．ｏｆＭａｔｈ．，ＤａｌｉｎＵｎｉｖ．ｏｆＳｃｉ．ａｎｄＴｅｃｈ．）Ａｂｓｔｒａｃｔ：...     

加密网格点二元局部基插值样条函数

1.简介 由于在理论以及应用两方面的重要性,多元样条引起了许多人的注意([6],[7]),紧支撑光滑分片多项式函数对于曲面的逼近是一个十分有效的工具。由于它们的局部支撑性,它们很容易求值;由于它们的光滑性,它们能被应用到要满足一定光滑条件的情况下;由于它们是紧支撑的,它们的线性包有很大的逼近灵活性,而且用它们构造逼近方法来解决的系统是     

Dimensions of spline spaces over T-meshes

A T-mesh is basically a rectangular grid that allows T-junctions. In this paper, we propose a method based on Bézier nets to calculate the dimension of a spline function space over a T-mesh. When the order of the smoothness is less than half of the degree of the spline functions, a dimension formula is derived which involves only the topological quantities of the T-mesh. The construction of basis functions is briefly discussed. Furthermore, the dimension formulae for T-meshes after mesh operations, such as edge insertion and mesh merging, are also obtained.     

Spline Surfaces over Arbitrary Topological Meshes: Theoretical Analysis and Application

Based on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to $C^2$-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate $C^1$-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational $C^1$-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The $C^1$−continuous connection schemes between two patches of the surfaces are presented.     

NEW PROOF OF DIMENSION FORMULA OF SPLINE SPACES OVER T-MESHES VIA SMOOTHING COFACTORS

A T-mesh is basically a rectangular grid that allows T-junctions. Recently, Deng etal introduced splines over T-meshes, which are generalizations of T-splines invented by Sederberg etal, and proposed a dimension formula based on the B-net method. In this paper, we derive an equivalent dimension formula in a different form with the smoothing cofactor method.     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

B-splines on 3-D tetrahedron partition in four-directional mesh

It is more difficult to construct 3-D splines than in 2-D case. Some results in the three directional meshes of bivariate case have been extended to 3-D case and corresponding tetrahedron partition has been constructed. The support of related B-splines and their recurrent formulas on integration and differentiation-difference are obtained. The results of this paper can be extended into higher dimension spaces, and can be also used in wavelet analysis, because of the relationship between spline and wavelets.     

The instability degree in the diemnsion of spaces of bivariate spline

In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of spaces of bivariate spline is presented. The results in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity.     

How to build all Chebyshevian spline spaces good for geometric design?

In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations.     

Bivariate C1 Cubic Spline Spaces Over Even Stratified Triangulations

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C ¹ cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C ¹ cubic spline spaces over a so-called even stratified triangulation.     

一类分层三角剖分下三次样条空间的维数

本文定义了平面单连通多边形域的一类较任意的三角剖分－分层三角剖分，并通过分析二元样条的积分协调条件，确定了分层三角剖分卜三次Ｃ^１作条函数空间的维数．     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

Embeddings of Beppo–Levi spaces in Hölder–Zygmund spaces, and a new method for radial basis function interpolation error estimates

The Beppo–Levi native spaces which arise when using polyharmonic splines to interpolate in many space dimensions are embedded in Hölder–Zygmund spaces. Convergence rates for radial basis function interpolation are inferred in some special cases.     

Minimal splines and wavelets

This paper is dedicated to the memory of the prominent mathematician S.G. Mikhlin. Here, Mikhlin’s idea of approximation relations is used for construction of wavelet resolution in the case of spline spaces of zero height. These approximation relations allow one to establish the embedding of the spline spaces corresponding to nested grids. Systems of functionals which are biorthogonal to the basic splines are constructed using the relations; then the systems obtained are used for wavelet decomposition. It is established that, for a fixed pair of grids of which one is embedded into the other and for an arbitrary fixed (on the coarse grid) spline space, there exists a continuum of spline spaces (on the fine grid) which contain the aforementioned spline space on the coarse grid. The wavelet decomposition of such embedding is given and the corresponding formulas of decomposition and formulas of reconstruction are deduced. The space of ( , φ)-splines is introduced with three objects: the full chain of vectors, prescribed infinite grid on real axis and the preassigned vector-function φ with m + 1 components (m is called the order of the splines). Under certain assumptions, the splines belong to the class C ^{m ? 1}. The gauge relations between the basic splines on the coarse grid and the basic splines on the fine grid are deduced. A general method for construction of a biorthogonal system of functionals (to basic spline system) is suggested. In this way, a chain of nested spline spaces is obtained, and the wavelet decomposition of the chain is discussed. The spaces and chains of spaces are completely classified in the terms of manifolds. The manifold of spaces considered is identified with the manifold of complete sequences of points of the direct product of an interval on the real axis and the projective space ? ^m ; the manifold of nested spaces is identified with the manifold of nested sequences of points of the direct product mentioned above.     

(3,3,1,1)层次B样条与分级T网格上多项式样条的相互转换

1引言 B样条在计算机图形学和几何建模等领域有着广泛的应用[3,8].在应用过程中,通常都需要对得到的模型进行修改以到达更好的效果.对于B样条曲线,利用节点插入算法可以有效地进行局部修改.     

On Spline Approximation for Singular Integral Equations on an Interval

This paper analyses the convergence of spline approximation methods for strongly elliptic singular integral equations on a finite interval. We consider collocation by smooth polynomial splines of odd degree multiplied by a weight function and a Galerkin-Petrov method with spline trial functions of even degree and piecewise constant test functions. We prove the stability of the methods in weighted Sobolev spaces and obtain the optimal orders of convergence in the case of graded meshes.     

On a general new class of quasi Chebyshevian splines

We prove that a general class of splines with sections in different Extended Chebyshev spaces or in different quasi Extended Chebyshev spaces can be viewed as quasi Chebyshevian splines, that is, as splines with all sections in a single convenient quasi Extended Chebyshev space. As a result, we can affirm the presence of blossoms in the corresponding spline spaces, with all the important consequences inherent in blossoms, namely, the possibility of developing all design algorithms for splines, the existence of B-splines bases, along with their optimality.