Macro-elements and stable local bases for splines on Powell-Sabin triangulations

Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundary-value problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.

     

Blossoms and Optimal Bases

It is now classical to define blossoms by means of intersections of osculating flats. We consider here the most general context of spline spaces with sections in arbitrary extended Chebyshev spaces and with connections defined by arbitrary lower triangular matrices with positive diagonal elements. We show how the existence of blossoms in such spaces automatically leads to optimal bases in the sense of Carnicer and Peña.     

Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

We describe a method which can be used to interpolate function values at a set of scattered points in a planar domain using bivariate polynomial splines of any prescribed smoothness. The method starts with an arbitrary given triangulation of the data points, and involves refining some of the triangles with Clough-Tocher splits. The construction of the interpolating splines requires some additional function values at selected points in the domain, but no derivatives are needed at any point. Given n data points and a corresponding initial triangulation, the interpolating spline can be computed in just O(n) operations. The interpolation method is local and stable, and provides optimal order approximation of smooth functions.     

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.     

Which spaces for design?

We determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms. It is the class of all spaces containing constants of which the spaces derived under differentiation are Quasi Extended Chebyshev spaces, i.e., they permit Hermite interpolation, Taylor interpolation excepted. It is also the class of all spaces which possess Bernstein bases, or of all spaces for which any associated spline space does possess a B-spline basis. Note that blossoms guarantee that such bases are normalised totally positive bases. They even are the optimal ones.     

On the Equivalence Between Existence of B-Spline Bases and Existence of Blossoms

In spline spaces with sections in arbitrary extended Chebyshev spaces and with connections defined by arbitrary lower triangular matrices with positive diagonal elements, we prove that existence of B-spline bases is equivalent to existence of blossoms. As is now classical, we construct blossoms with the help of osculating flats. As for B-spline bases, this expression denotes normalized basis consisting of minimally supported functions which are positive on the interior of their supports and which satisfy an additional end point condition.     

Sequence spaces of spline functions on subsets and l-spaces

Spaces of sequences of bounded linear splines defined on arbitrary subsets of are studied, especially with respect to continuous extensions. An extension problem is solved by establishing a decomposition for the space of spline sequences with respect to the l^∞-space on a corresponding subset of . An application to Zygmund spaces on subsets is presented.     

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.     

C 1 Quintic Splines on Type-4 Tetrahedral Partitions

Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type-4) partition and associated trivariate C ¹ quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is also shown how the spaces can be used to solve certain Hermite and Lagrange interpolation problems.     

B-spline bases for unequally smooth quadratic spline spaces on non-uniform criss-cross triangulations

In this paper, we investigate bivariate quadratic spline spaces on non-uniform criss-cross triangulations of a bounded domain with unequal smoothness across inner grid lines. We provide the dimension of the above spaces and we construct their local bases. Moreover, we propose a computational procedure to get such bases. Finally we introduce spline spaces with unequal smoothness also across oblique mesh segments.     

A Lower Bound for the Dimension of Trivariate Spline Spaces

We consider trivariate C^r spline spaces of degree d defined on arbitrary tetrahedral partitions. A lower bound for the dimension of trivariate spline spaces over arbitrary tetrahedral partitions for d > r is computed. This is the first general lower bound known.     

A Method Which Generates Splines in H-Locally Convex Spaces and Connections with Vectorial Optimization

In this research paper we present a modality for generating splines in H-locally convex spaces which allows us to solve some problems of best approximation by linear subspaces of spline functions in these spaces. In this way one shows that the elements of best vectorial approximation coincide with the spline functions introduced by us in a previous research work. These splines are also the only elements of best simultaneous approximation by their generated linear subspaces with respect to any family of seminorms which induces the H-locally convex topology and, consequently, they are the only solutions for some frequent strong and vectorial optimization programs. Moreover, as we shall see in the numerical examples, our construction leads to discover orthogonal decompositions for H-locally convex spaces which, in general, are difficult to be identified.     

散乱数据的多项式自然样条光顺与广义插值

由于理论与实践的重要性,在多元插值方面有相当多的工作,如[1]-[11]。目前以箱样条(box splines),光滑余因子与B网方法以及薄板样条与径函数(radial basis function)方法比较活跃。前者具有良好的性质和丰富的结构,很快成为一个活跃的研究方向,最近更在小波(wavelet)变换理论研究上发挥了作用。但是,它一般只处理规则分划的问题,不能做多元散乱数据的插值。     

Asymptotically optimal approximation in fractional Sobolev spaces and the numerical solution of differential equations

We study two extremal problems in fractional Sobolev spaces which were motivated by the consideration of optimal finite difference methods for approximating the solution of a well posed Cauchy problem for linear partial differential equations. The limiting forms as the mesh increment tends to zero of the solutions of the extremal problem are shown to be generalized spline functions. Computations illustrating the effectiveness of these generalized spline functions are included.This research was partially supported by ONR Contract N 0014-69-C-0023.     

A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations

In the paper, a family of bivariate super spline spaces of arbitrary degree defined on a triangulation with Powell–Sabin refinement is introduced. It includes known spaces of arbitrary smoothness r and degree \(3r-1\) but provides also other choices of spline degree for the same r which, in particular, generalize a known space of \(\mathscr {C}^{1}\) cubic super splines. Minimal determining sets of the proposed super spline spaces of arbitrary degree are presented, and the interpolation problems that uniquely specify their elements are provided. Furthermore, a normalized representation of the discussed splines is considered. It is based on the definition of basis functions that have local supports, are nonnegative, and form a partition of unity. The basis functions share numerous similarities with classical univariate B-splines.     

Discrete orthogonal projections on multiple knot periodic splines

This paper establishes properties of discrete orthogonal projections on periodic spline spaces of order r, with knots that are equally spaced and of arbitrary multiplicity Mr. The discrete orthogonal projection is expressed in terms of a quadrature rule formed by mapping a fixed J-point rule to each sub-interval. The results include stability with respect to discrete and continuous norms, convergence, commutator and superapproximation properties. A key role is played by a novel basis for the spline space of multiplicity M, which reduces to a familiar basis when M=1.     

Spline prewavelets for non-uniform knots

Summary In this paper, we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non-uniform knot-sequences. To this end, we show, among other things, the existence of compactly supported prewavelets and of prewavelets that are globally supported, but decay exponentially. In each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets.     

样条空间S~μ_k（△）的维数

本文考虑了欧式空间Ｒ￣ｎ中任意单纯形剖分上的样条函数空间．证明了当ｋ≥（３μ＋１）２￣（ｎ－２）＋１时，计算任意单纯形剖分Δ上的ｋ次μ阶光滑样条空间的维数，可归结为计算每个σ－关联域（ｉ－单纯形σ∈Δ）Ｒ（σ）上的２￣（ｎ－ｉ－１）μ次μ阶光滑（ｉ≤ｎ－１）样条空间的维数。这里σ－关联域Ｒ（σ）是指Δ中所有包含σ的单纯形所成的单纯形剖分．     

On Almost Interpolation

We obtain some characterizations of almost interpolation configurations of points with respect to finite-dimensional functional spaces. Particularly, a Schoenberg–Whitney type characterization which is valid for any multivariate spline space relative to an arbitrary partition of a domainA⊂^mis presented. As a closely related problem we investigate sectional structure of finite-dimensional spaces of real functions on a topological spaceA. It is shown that under some reasonable restrictions onAany space of this sort may be considered as piecewise almost Chebyshev.     

Hierarchical Riesz Bases for Hs(Ω), 1 < s < 5/2

On arbitrary polygonal domains $\Omega \subset \RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(\Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s \in (2,\frac{5}{2})$ to $s \in (1,\frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned.