Three-monotone spline approximation

For r≥3, n∈N and each 3-monotone continuous function f on [a,b] (i.e., f is such that its third divided differences [x₀,x₁,x₂,x₃]f are nonnegative for all choices of distinct points x₀,…,x₃ in [a,b]), we construct a spline s of degree r and of minimal defect (i.e., s∈C^r−1[a,b]) with n−1 equidistant knots in (a,b), which is also 3-monotone and satisfies ‖f−s‖_L∞[a,b]≤cω₄(f,n⁻¹,[a,b])_∞, where ω₄(f,t,[a,b])_∞ is the (usual) fourth modulus of smoothness of f in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots.Moreover, we also prove a similar estimate in terms of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L_p norm with p<∞. At the same time, positive results in the L_p case with p<∞ are still valid if one allows the knots of the approximating ppf to depend on f while still being controlled.These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are “positive”) and k-monotone approximation with k≥4 (where just about everything is “negative”).     

New spline basis functions for sampling approximations

We describe a method of constructing a new kind of splines with compact support on . These basis functions consisting of a linear combination of the cardinal B-splines of mixed orders enable us to achieve simultaneously a good sampling approximation and an interpolation of any smooth function.     

Nonuniqueness and selections in spline approximation

This paper studies problems of nonuniqueness for the metric projection ofC(T),T a compact Hausdorff space, onto a finite-dimensional subspaceG, and discusses the results for polynomial spline approximation. Among others, we prove that the metric projection ofC[a, b] ontoS _k,n , the space of polynomial splines of degree less than or equal ton withk simple knots in (a, b), is lower semicontinuous on an open, dense subset ofC[a, b] and, consequently, any standard selection of the projection is continuous on this subset. We further show that continuous selections are not so easy to construct.Communicated by Ronald A. DeVore.     

An algorithm for determining the approximation orders of multivariate periodic spline spaces

We give an algorithm which computes the approximation order of spaces of periodic piece-wise polynomial functions, given the degree, the smoothness and tesselation. The algorithm consists of two steps. The first gives an upper bound and the second a lower bound on the approximation order. In all known cases the two bounds coincide.     

Bivariate segment approximation and splines

The problem to determine partitions of a given rectangle which are optimal for segment approximation (e.g., by bivariate piecewise polynomials) is investigated. We give criteria for optimal partitions and develop algorithms for computing optimal partitions of certain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain good partitions for interpolation by tensor product spline spaces. Our numerical examples show that the methods work efficiently.     

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines S_q^r(Δ¹) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.     

Numerical integration over polygons using an eight-node quadrilateral spline finite element

In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures.     

Extrema detection of bivariate spline functions

In this paper, we develop a systematic method for detecting the extrema of bivariate spline functions and of their derivatives. It is assumed that the splines are constituted by employing normalized, uniform B-splines as the basis functions, and the detection procedure fully utilizes the spline properties. All the extrema can be found except those with singular Hessian matrix. By numerical examples, we demonstrate the effectiveness of the method.     

Modified Taylor approximation of functions with periodic behaviour

We approximate a function with periodic behaviour by means of a small modification of its Taylor polynomial. This modification is based on the work of Scheifele and will simplify the construction of special numerical methods for differential equations with near periodic solutions.     

Free-knot spline approximation of stochastic processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the s-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average L_p-distance to the splines spaces, as the (expected) number of free knots tends to infinity.     

An analysis of cubic approximation schemes for conic sections

In this paper a piece of a conic section is approximated by a cubic or piecewise cubic polynomial. The main tool is to define the two inner control points of the cubic as an affine combination, defined by [0, 1], of two control points of the conic. If is taken to depend on the weightw of the latter, a function (w) results which is used to distinguish between different algorithms and to analyze their properties. One of the approximations is a piecewise cubic havingG ⁴ continuity at the break points.     

Uniform approximation of min/max functions by smooth splines

In some optimization problems, min(f₁,…,f_n) usually appears as the objective or in the constraint. These optimization problems are typically non-smooth, and so are beyond the domain of smooth optimization algorithms. In this paper, we construct smooth splines to approximate min(f₁,…,f_n) uniformly so that such optimization problems can be dealt with as smooth ones.     

Maximal approximation order for a box-spline semi-cardinal interpolation scheme on the three-direction mesh

Let M be the centred 3-direction box-spline whose direction matrix has every multiplicity 2. A new scheme is proposed for interpolation at the vertices of a semi-plane lattice from a subspace of the cardinal box-spline space generated by M. The elements of this semi-cardinal box-spline subspace satisfy certain boundary conditions extending the not-a-knot end-conditions of univariate cubic spline interpolation. It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order 4. AMS subject classification 41A15, 41A05, 41A25, 41A63, 39A70, 47B35     

A linear approach to shape preserving spline approximation

This paper deals with the approximation of a given large scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data.Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local linear sufficient conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the resulting minimisation problem is also linear, as the problem can then be written as a linear programming problem. A special linear approach based on constrained least squares is presented, which in the case of large data reduces the complexity of the problem sets in contrast with that obtained for the usual ₂-norm as well as the -norm.An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.     

Perfect Category-Graded Algebras

In a perfect category, every object has a minimal projective resolution. We give a criterion for the category of modules over a category-graded algebra to be perfect.     

Approximation properties of the spline fit

Different topics in connection with spline fit are discussed in this paper. In particular, an example is given showing non-convergence of splines, and further some error bounds of cubic spline interpolation are proved.     

THE DIMENSION OF A CLASS OF BIVARIATE SPLINE SPACES

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Bounds on the dimensions of trivariate spline spaces

We derive upper and lower bounds on the dimensions of trivariate spline spaces defined on tetrahedral partitions. The results hold for general partitions, and for all degrees of smoothness r and polynomial degrees d.      

3-正则图的不共边的完美匹配（英文）

研究3-正则图的一个有意义的问题是它是否存在k个没有共边的完美匹配.关于这个问题有一个著名的Fan-Raspaud猜想：每一个无割边的3-正则图都有3个没有共边的完美匹配.但这个猜想至今仍未解决.设dim（P（G））表示图G的完美匹配多面体的维数.本文证明了对于无割边的3-正则图G,如果dim（P（G））≤14,那么k≤4：如果dim（P（G））≤20,那么k≤5.