On the problems of smoothing and near-interpolation

In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing.

     

Banach lattices which are order uniformly noncreasy

The class of order uniformly noncreasy Banach lattices is introduced and studied. A fixed point theorem for those lattices is proved.     

Local smoothing estimates for some half-wave operators

In this paper we study the local smoothing estimates for some half-wave operators associated to nondegenerate convex hypersurfaces.     

An estimate for the least uniform deviation from the class of functions with bounded second derivative

We consider the problem of uniform approximation of a continuous function on an interval by means of the class of functions with bounded second derivative. We prove an estimate for the least uniform deviation of a function in terms of its local deviations on uniform and nonuniform three-point grids.     

Local approximation of uniformly regular Carnot-Carathéodory quasispaces by their tangent cones

We prove a local approximation theorem for the Carnot-Carathéodory quasimetrics on uniformly regular (equiregular) Carnot-Carathéodory spaces. Using this theorem, we study convergence of the Carnot-Carathéodory quasispaces to their tangent cones. In particular, we prove a Mitchell type theorem on convergence of an equiregular Carnot-Carathéodory quasispace with distinguished point to its tangent cone.     

M-Cross-Validation in Local Median Estimation

M-cross-validation criterion is proposed for selecting a smoothing parameter in a nonparametric median regression model in which a uniform weak convergency rate for the M-cross-validated local median estimate, and the upper and lower bounds of the smoothing parameter selected by the proposed criterion are established. The main contribution of this study shows a drastic difference from those encountered in the classical L2-, L1- cross-validation technique, which leads only to the consistency in the sense of the average. Obviously, our results are novel and nontrivial from the point of view of mathematics and statistics, which provides insight and possibility for practitioners substituting maximum deviation for average deviation to evaluate the performance of the data-driven technique.     

Sharpness of estimates for the least uniform derivation from the class of functions with bounded second derivative

We study the uniform approximation of continuous functions on an interval by functions with bounded second derivative. We prove the sharpness of estimates for the least uniform deviation of a function in terms of its local deviations on uniform and nonuniform three-point grids.     

关于Bernstein算子的收敛速度

本文对一类函数建立了Bernstein算子的一致逼近定理,而且给出了其逆定理的一个简短证明。     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

The smoothing Newton method for solving a system of nonsmooth equations , which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the th step, the nonsmooth function is approximated by a smooth function , and the derivative of at is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if is semismooth at the solution and satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is -uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

     

弱紧局部一致凸空间中的度量投影

本文研究了弱紧局部一致凸空间中度量投影的弱连续性,将Ｂ．Ｂ．Ｐａｎｄａ和Ｏ．Ｐ．Ｋａｐｏｏｒ的相应结论推广到更一般的弱紧局部一致凸空间．最后给出了局部一致凸点的一个必要条件．     

Weighted Best Local Approximation

In this survey, the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the Lpspaces and recent results in Orlicz spaces. The notion of balanced point, which was introduced by Chui et al. in 1984 are extensively used.     

关于Baskakov－Kurrmeyer算子的一致逼近

本文首先给出了Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子在一致副近意义下的正定理，并把它推广到一类线性组合的情形，然后讨论了它的导数与光滑模的等价关系，最后给出了二元Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子逼近阶的特征刻画．     

关于紧集上局部Lipschitz条件的讨论

许多常微分方程教材关于解的整体连续依赖性的讨论都用到了一个紧性事实:欧氏空间中的紧集上一个局部Lipschitz函数一定在该紧集上是全局Lipschitz的.然而这一事实在教学中并非显然,不少学生在试图给出证明时都走入了一个误区.本文对这一问题从正反两方面进行了讨论.     

Global smoothness and uniform convergence of smooth Poisson-Cauchy type singular operators

In this article we introduce the smooth Poisson-Cauchy type singular integral operators over the real line. Here we study their simultaneous global smoothness preservation property with respect to the L_p norm, 1?p?∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

Local Connectedness Made Uniform

The uniformly locally connected reflection for a locally connected uniform frame is constructed. Applications of this construction to the theory of locally connected completely regular frames are given. One such application is that if a completely regular frame is locally connected and pseudocompact then every compactification of it is locally connected.     

Cp Condition and the Best Local Approximation

In this paper, we introduce a condition weaker than the Lpdifferentiability,which we call Cpcondition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be Lpdifferentiable. In addition, we study the convexity of the set of cluster points of the net of best appoximations of f,{Pe( f)} as e → 0.     

基于二次函数光滑化逼近的修正低阶罚函数

针对不等式约束优化问题, 给出了通过二次函数对低阶精确罚函数进行光滑化逼近的两种函数形式, 得到修正的光滑罚函数. 证明了在一定条件下, 当罚参数充分大, 修正的光滑罚问题的全局最优解是原优化问题的全局最优解. 给出的两个数值例子说明了所提出的光滑化方法的有效性.     

Local structure theorems for smooth maps of formal schemes

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007) 1341-1367]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism.