关于三角剖分下二元样条空间的维数

本文利用多元样条的协调条件、平面图理论和空间同构的方法建立了对于所有μ≥1,k≥4μ+1和任意三角剖分T的样条空间S_k^μ(T)的维数公式,对于一类尽可能任意的三角剖分T=T(μ),本文也给出了当k≥μ+1时S_k^μ(T)的维数公式。特别当μ=1,2时此类剖分在逼近问题上有明显的实用性。     

拟贯穿剖分下多元样条空间基

本文给出了拟贯穿剖分△_qc下的平面区域D上的样条函数空间S_k^u(△_qc,D)的基底。     

二元四次样条函数空间的维数与基底

本文讨论了一类多边形区域上样条空间S₄²(D,△)的维数与基底,给出并证明了文献[1]中主要结果的推广形式。 在文献[1]中,作者解决了平面矩形区域上样条函数空间S_k^μ(△_mn⁽¹⁾)(k=3,μ=1;k=4,μ=2)的基底问题,其主要结果是基本的。本文将要考虑其中有关S₄²(△_mn⁽¹⁾     

多元弱样条函数空间的维数

讨论了多元弱样条一点处的维数公式及任意三角剖分下的维数公式．得到了１－型剖分下Ｗ_３^１（Ｉ₁Δ_ｍｎ^１）的维数与局部支集样条基．     

关于二元样条空间Sr（3r）（△*）的维数

设△^*任何三角剖分△的ＨＣＴ细分的三角剖分．本文建立了定义于△^*上的二元样条函数空间S^r_（3r）（△^*）的维数公式．我们的证明方法同时给出了S^r_（3r）（△^*）的一组显示的基函数，并阐明基函数具有某种意义的局部最小支集     

二元三次样条空间S31,2(Δmn(2)) 的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2 型三角剖分上二元三次样条空间S₃^1,2(Δ_mn⁽²⁾)的若干样条拟插值算子. 这些变差缩减算子由样条函数B_ij¹支集上5 个网格点或中心和样条函数B_ij²支集上5 个网格点处函数值定义. 这些样条拟插值算子具有较好的逼近性,甚至算子V_mn（f） 能保持近最优的三次多项式性. 然后利用连续模,分析样条拟插值算子V_mn（f）一致逼近于充分光滑的实函数. 最后推导误差估计.     

关于一个二元B样条基

连接矩形网剖分中每一矩形的两条对角线得到一个三角剖分,将它记为△_mn。当k≥3时,△_mn上不存在k—1阶光滑度的分片k次非平凡局部支集二元样条函数,所以本文给出了均匀剖分下的具有最小对称支集的二元二次一阶光滑度的B样条基。此外,作为一元样条的Marsden恒等式的推广,我们还得到了二元样条的相应形式以及其它一些恒等式。利用这些恒等式,我们在整个剖分△_mn的二次C¹样条函数空间上建立逼近误差估计以及相应的渐近公式。     

关于一类S1，13（△（2）mn）插值与逼近

设△^（2）_mn是矩形域Ｄ＝［ａ，ｂ］(?)［ｃ，ｄ］的Ⅱ－型三角剖分．S^1，1₃（△^（2）_mn）是带边界条件的二元三次样条空间：本文我们将讨论一类S^1，1₃（△^（2）_mn）的插值问题，证明了它的存在性，唯一性及逼近阶：如果ｆ∈Ｃ^４（Ｄ），则有｜ｆ－ｓ｜≤ｋ（ｌ）·ｍａ     

一类Ⅱ型剖分下的二元三次周期样条的超限插值和逼近(英)

本文讨论了Ⅱ一型三角剖分△^（2）_mn下的一类二元三次周期样条的超限插值和逼近,给出了它的表示以及存在唯一性,最后,估计了它的逼近阶．     

多元 Besov-Wiener 类的无穷维宽度和最优恢复

该文考虑 Besov-Wiener 类S_pqθ^r B(R^d)和 S_pqθ^r B(R^d)在 L_q(R^d) 空间下 (1≤q≤ p <∞ ) 的无穷维σ -宽度和最优恢复问题.通过考虑样条函数逼近和构造一种连续样条算子, 得到了关于无穷维Kolmogorov 宽度、无穷维线性宽度、无穷维 Gel'fand 宽度和最优恢复的弱渐近结果.     

Network Splines

A multivariate interpolant to scattered data is developed by generalizing (weighted) bivariate network splines to an n-dimensional setting. A graph joining the data points serves to define a set of edges over which an interpolating curve network is constructed subject to smoothness and minimal energy constraints. A subsequent extension of the curve network to the convex hull of the data points defines a smooth interpolating surface. The problems of existence and uniqueness are investigated and some examples of interpolants to rapidly varying data in ?³ and ?⁴ are presented.     

Multi-Box Splines

We construct local generators, comprising r functions, for refinable spaces of bivariate C^n-1 spline functions of degree n on meshes comprising all lines through points of the integer lattice in the directions of n + r + 1 pairwise linearly independent vectors with integer components. The generators are characterised by their Fourier transforms. Their shifts are shown to form a Riesz basis if and only if at most r lines in the mesh intersect other than in the integer lattice, which can occur for n ≤ 2r - 1. The symmetry of these generators is studied and examples are given.     

Martensen Splines

This paper is concerned with the construction of the fundamental functions associated with a two-point Hermite spline interpolation scheme used by Martensen in the context of the remainder of the Gregory quadrature rule. We derive both a recursive construction and an explicit representation in terms of the underlying B-Splines which can easily be deduced using Marsden’s identity. We can make use of these functions in order to introduce a local interpolation scheme which reproduces all splines. Finally, we examine the error of this interpolant to a sufficiently smooth function and realize that it behaves like in the case of splines of degree n. AMS subject classification (2000) 65D05, 65D07, 41A15     

Riesz Bases of Splines and Regularized Splines with Multiple Knots

This paper deals withL₂(R)-norm and Sobolev-norm stability of polynomial splines with multiple knots, and with regularized versions thereof. An essential ingredient is a result on Hölder continuity of the shift operator operating on a B-spline series. The stability estimates can be reformulated in terms of a Riesz basis property for the underlying spline spaces. These can also be employed to derive a result on stable Hermite interpolation on the real line. We point to the connection with the problem of symmetric preconditioning of bi-infinite interpolation matrices.     

Adaptive Free-Knot Splines

This article proposes a function estimation procedure using free-knot splines as well as an associated algorithm for implementation in nonparametric regression. In contrast to conventional splines with knots confined to distinct design points, the splines allow selection of knot numbers and replacement of knots at any location and repeated knots at the same location. This exibility leads to an adaptive spline estimator that adapts any function with inhomogeneous smoothness, including discontinuity, which substantially improves the representation power of splines. Due to uses of a large class of spline functions, knot selection becomes extremely important. The existing knot selection schemes—such as stepwise selection—suffer the difficulty of knot confounding and are unsuitable for our purpose. A new knot selection scheme is proposed using an evolutionary Monte Carlo algorithm and an adaptive model selection criterion. The evolutionary algorithm locates the optimal knots accurately, whereas the adaptive model selection strategy guards against the selection error in searching through a large candidate knot space. The performance of the procedure is examined and illustrated via simulations. The procedure provides a significant improvement in performance over the other competing adaptive methods proposed in the literature. Finally, usefulness of the procedure is illustrated by an application to actual dataset.     

Semilocal Smoothing Splines

The paper is aimed at periodic and nonperiodic semilocal smoothing splines, or S-splines of class C p, formed by polynomials of degree n. The first p?+?1 coefficients of each polynomial are determined by the values of the preceding polynomial and its first p derivatives at the glue-points, while the remaining n???p coefficients of the higher derivatives of the polynomial are found by the method of least squares. These conditions are supplemented with the initial conditions (nonperiodic case) or the periodicity condition on the spline-function on the segment where it is defined. A linear system of equations is obtained for the coefficients of the polynomials constituting the spline. Its matrix has a block structure. Existence and uniqueness theorems are proved and it is shown that that the convergence of the splines to the original function depends on the eigenvalues of the stability matrix. Examples of stable S-splines are given.     

On Mixed Interpolating Splines

Suppose that n is odd and 0=x_0相似文献   

Splines and biorthogonal systems

The paper constructs coordinate splines on a closed interval, provides realizations of the corresponding biorthogonal system, and constructs finite-dimensional spaces of splines (nonpolynomialin general) of the class C ¹. Bibliography: 7 titles.     

Classification of Refinable Splines

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with noninteger dilations and arbitrary translations is studied in this paper. We classify completely all refinable splines with integer translations and arbitrary dilations. Our study involves techniques from number theory and complex analysis.