Interpolation Operators on the Space of Holomorphic Functions on the Unit Circle

The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval [—a, a],a (0,1), have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general extremal problem with free weights and knots is solved.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

Hermite interpolation of nonsmooth functions preserving boundary conditions

This article is devoted to the construction of a Hermite-type regularization operator transforming functions that are not necessarily into globally finite-element functions that are piecewise polynomials. This regularization operator is a projection, it preserves appropriate first and second order polynomial traces, and it has approximation properties of optimal order. As an illustration, it is used to discretize a nonhomogeneous Navier-Stokes problem, with tangential boundary condition.

     

Hermite interpolation by Pythagorean hodograph curves of degree seven

Polynomial Pythagorean hodograph (PH) curves form a remarkable subclass of polynomial parametric curves; they are distinguished by having a polynomial arc length function and rational offsets (parallel curves). Many related references can be found in the article by Farouki and Neff on Hermite interpolation with PH quintics. We extend the Hermite interpolation scheme by taking additional curvature information at the segment boundaries into account. As a result we obtain a new construction of curvature continuous polynomial PH spline curves. We discuss Hermite interpolation of boundary data (points, first derivatives, and curvatures) with PH curves of degree 7. It is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials. With the help of the canonical Taylor expansion of planar curves, we analyze the existence and shape of the solutions. More precisely, for Hermite data which are taken from an analytical curve, we study the behaviour of the solutions for decreasing stepsize . It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize , provided that certain technical assumptions are satisfied. Moreover, this solution matches the shape of the original curve; the approximation order is 6. As a consequence, any given curve, which is assumed to be (curvature continuous) and to consist of analytical segments can approximately be converted into polynomial PH form. The latter assumption is automatically satisfied by the standard curve representations of Computer Aided Geometric Design, such as Bézier or B-spline curves. The conversion procedure acts locally, without any need for solving a global system of equations. It produces polynomial PH spline curves of degree 7.

     

Convergence of Runge-Kutta Methods for Delay Differential Equations

This paper deals with the adaptation of Runge—Kutta methods to the numerical solution of nonstiff initial value problems for delay differential equations. We consider the interpolation procedure that was proposed in In 't Hout [8], and prove the new and positive result that for any given Runge—Kutta method its adaptation to delay differential equations by means of this interpolation procedure has an order of convergence equal to min {p,q}, where p denotes the order of consistency of the Runge—Kutta method and q is the number of support points of the interpolation procedure.This revised version was published online in October 2005 with corrections to the Cover Date.     

MODIFIABLE QUARTIC AND QUINTIC CURVES WITH SHAPE-PARAMETERS

1　IntroductionBecause of their good properties,the cubic Bézier,B-spline and NURBScurves play animportantrole in CAD,CAGD and modeling systems.When interpolation by the abovecurvesto all ora partofthe control pointsisrequired,itis necessary eitherto find new control pointsby solving a system of linear equations or to insert additional control points. Moreover,thewhole interpolating curve may be affected by moving an individual control point[1～ 6] .By uisng the matrix form ofthe Bernst…     

On an Extremal Scale of Approximation Spaces

This paper is concerned with an extremal scale of approximation spaces involving logarithmic weights. We give equivalent norms and study the behavior of this scale under reiteration and interpolation. Some applications to Lorentz-Zygmund operator ideals and Besov spaces with logarithmic weights are given.     

激光共焦扫描光学显微镜系统中的图像层间插值技术

由断层图像恢复出目标物体的三维图像是激光共焦扫描光学显微镜 (LCSM)系统的重要部分。提出了基于距离变换的断层图像插值方法来获得三维重构时的数据场。此方法由于在插值时考虑了目标物体的整体特征 ,从而使插值具有连续性     

Polynomial interpolation in several variables

This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included. This revised version was published online in June 2006 with corrections to the Cover Date.     

自由曲面形状信息快速反求系统研究

本文对自由曲面的快速反求系统进行了研究。首先将矩形光栅投影在被测表面上,利用光栅投影法测量自由曲面上每点高度信息。然后对测量的三维离散数据进行一种特殊的Hermite插值,通过重叠的矩形作用域叠加而形成一个由矩形网格表示的新曲面。最后,采用AutoCAD命令绘制三维自由曲面。