Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.     

Asymptotics for the remainder of a class of positive quadratures for integrands with an interior singularity

Summary For functions with an interior singularity, the errors of a class of positive quadrature formulae with high algebraic degree are reduced to those of the much simpler Euler-Maclaurin type formulae. Applying this method to certain classes of functions, such as, for example,f(x)=h(x)|x-u| , where >–1, with a sufficiently smooth functionh, we obtain the main term of the error expansion for quadrature rules of ultraspherical type.     

含铜MCM—41型分子筛的合成与表征

本文首次合成出一种具有MCM-41型结构的中孔（mesoporous）CuMCM-41分子筛，通过XRD、骨架IR、紫外漫反射、电子自旋共振、固体魔角核磁、TG—DTA等测试结果，表征铜（Ⅱ）在分子筛中四面体配位和八面体配位形式共存。     

Nitrogen Adsorption Study of MCM-41 Molecular Sieves Synthesized Using Hydrothermal Restructuring

Nitrogen desorption scanning hysteresis loops (DSHLs) for large-pore MCM-41 silicas (pore diameter from 4.0 to 6.5 nm) are reported for the first time. DSHLs for MCM-41 were compared with those of conventional mesoporous silicas and no appreciable differences were found, although hysteresis loops and DSHLs for the latter were usually broader. Since desorption behavior of conventional porous silicas is appreciably influenced by pore connectivity, the observed similarity in hysteresis behavior suggests single-pore blocking effects for MCM-41 due to variation of pore diameter along its nonintersecting channels. It was also shown that the steepness of nitrogen desorption branches at relative pressures close to 0.4 often results from proximity of the lower pressure limit of adsorption-desorption irreversibility and consequently it is not justified to consider it as an indication of narrow pore size distribution. Thus, application of desorption data in calculations of pore size distributions may be grossly misleading.     

Zr-MCM-41的合成及其表征

以Zr(NO3)4·3H2O为锆源，通过直接合成法得到了Zr-MCM-41介孔材料，考察了不同Zr/Si摩尔比对样品的影响，其中Zr/Si的摩尔比最大为0.25。随着Zr/Si的摩尔比的增加，2θ=2°附近的100晶面的衍射峰强度逐渐降低，当Zr/Si摩尔比大于0.25后得不到介孔Zr-MCM-41材料。对所得样品采用XRD、TG-DTA、N2吸附 脱附几种分析方法进行了表征。     

四甲基锡在MCM-41分子筛表面的接枝反应及产物性能

研究了真空条件下SnMe4在MCM-41分子筛表面的接枝反应，并用元素分析、ICP、GC-MS、XRD、FTIR、DRS、13C及119Sn MAS NMR、XPS、BET、TPD、TPR等方法对产物的组成、结构和性质进行了表征.结果表明, 两者可以定量地进行化学反应，将确定数目的甲基锡基团接枝在分子筛表面, 形成SnMe3/MCM-41物种；接枝反应发生在分子筛表面上, 不破坏MCM-41分子筛骨架结构；改性分子筛的BET比表面积有所降低，孔体积变小, 表面性质发生变化.四甲基锡在MCM-41上接枝反应的温度为343 K,比它与HY沸石的反应温度(193 K)高得多，并且产物SnMe3/MCM-41的热稳定性也高于SnMe3/HY.     

On constrained multivariate splines and their approximations

Summary The variational formulation of multivariate spline functions is generalized to include cases where the function has to satisfy inequality constraints such as positivity and convexity. Condition for existence and uniqueness of a solution is given. Approximation to the solution can be obtained by solving the variational problem in a finite dimensional subspace. Conditions for convergence and error estimates of the approximations are presented, both for interpolation problems and smoothing problems. The general theory is illustrated by specific examples including the volume-matching problem and the one-sided thin plate spline.This research is partially supported by the U.S. Army Contract No. DAAG 29-77-G-0207, and by NSF Grant No. MCS-8101836Part of this paper is based on Chapters 2 and 3 of the author's Ph. D. thesis. The author would like to express his sincere thanks to Professor Grance Wahba and to two referees for many helpful comments     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].