BestL 1 approximation by convex functions

It is shown that a function inL ₁ has a best approximation by convex functions, and that the net of bestL _p approximations converges asp decreases to one.     

Unicity of best one-sidedL 1-approximations

Summary This paper deals with the problem of uniqueness in one-sidedL ₁-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL ₁-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.     

Interpolatory properties of best rationalL 1-approximations

In this paper the distribution of the zeros of the error function for bestL ₁-approximation by rational functions fromR _n,m is considered. It is shown that the maximal distance between such zeros isO(1/(n–m)), ifn > m.Communicated by Edward B. Saff.     

Stability ofL 1-approximations and robustness of least module estimates

The paper is concerned with the robustness properties of least module (LM) estimates of regression parameters depending on the stability properties of L₁-norm approximations. The conditions of stability for L₁-approximations are obtained in linear, nonlinear, continuous, and discrete cases. The breakdown points of LM estimates and their upper bounds are derived in the linear case of approximation by algebraic and trigonometric polynomials, and in the nonlinear case of approximation by exponentials. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

二元C1四次样条插值

本文在较一般的平面三角剖分激造了一种C¹四次样条插值格式.这种格式仅用到被插函数的函数值与一阶导数值信息,并得出插值样条的递推计算格式.     

Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ɛ is c ₁ɛ^−1/4 + O(1), if the spline consists of parabolic arcs, and c ₂ɛ^−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c ₁ and c ₂ are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline. The research of SG and GV was partially supported by grant 6413 of the European Commission to the IST-2002 FET-Open project Algorithms for Complex Shapes in the Sixth Framework Program.     

Lagrange Interpolation by C 1 Cubic Splines on Triangulated Quadrangulations

We describe local Lagrange interpolation methods based on C ¹ cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.     

二元线性样条函数插值

二元样条函数插值在计算几何与计算机辅助几何设计中有着重要的作用.本文给出了一种矩形剖分上二元线性样条函数进行Lagrange插值时插值适定结点组所满足的拓扑与几何性质,这种性质依赖于二元线性样条函数所决定的分片线性代数曲线.     

On the bestL 1-approximations of functional classes by splines under restrictions imposed on their derivatives

We find the exact asymptotics (asn→∞) of the bestL ₁-approximations of classesW ₁ ^r of periodic functions by spliness∈S _{2n, r∼-1} (S _{2n, r∼-1} is a set of 2π-periodic polynomial splines of orderr−1, defect one, and with nodes at the pointskπ/n,k∈ℤ) such that V ₀ ^2π s( ^r-1)≤1+ɛ _n , where {ɛ _n } _n=1 ^∞ is a decreasing sequence of positive numbers such that ɛ _n n ²→∞ and ɛ _n →0 asn→∞. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 435–444, April, 1999.     

Extistence of best nonlinear Lp -approximations

We are concerned with the Tschebyscheff and L_p-approximation problem, when the approximating family is the union of solutions of linear differential equations. We prove a strong compactness property and a global existence theorem. Application to generalized exponential approximation is given.     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Calculation of linear bestL p -approximations

In this paper we are concerned with finding theL _p -solution (i.e. minimizing theL _p -norm of the residual vector) to a linear approximation problem or, equivalenty, to an overdetermined system of linear equations. An embedding method is described in which the damped Newton iteration is applied to a series of perturbed problems in order to guarantee convergence and also increase the convergence rate.     

Best Approximation by Free Knot Splines

We consider the problem of finding the best (uniform) approximation of a given continuous function by spline functions with free knots. Our approach can be sketched as follows. By using the Gauß transform with arbitrary positive real parameter t, we map the set of splines under consideration onto a function space, which is arbitrarily close to the spline set, but satisfies the local Haar condition and also possesses other nice structural properties. This enables us to give necessary and sufficient conditions for best approximations (in terms of alternants) and, under some assumptions, even full characterizations and a uniqueness result. By letting t 0, we recover best approximation in the original spline space. Our results are illustrated by some numerical examples, which show in particular the nice alternation behavior of the error function.     

Unicity of best one-sided L1-approximations for certain classes of spline functions

The chief purpose of this paper is to present conditions ensuring uniqueness of best one-sided L¹-approximations for approximation by finite dimensional subspaces of differentiable functions. Using these results it is shown that uniqueness of such best approximations will hold for a class of generalized spline subspaces and also for spline subspaces satisfying certain boundary conditions. These considerations have an important application to uniqueness of quadrature formulae of “highest possible degree of precision”.