Shape Preserving Interpolation to Convex Data

1.IntroductionA fundamental problem in computer graphics is the drawing of a smooth curve through aset of data points(xi,fi) (i=0 ,1 ,… ,n) .In many applications,particularly in scientificvisualisation,the y- values are depenenton the x- values and it is…     

Shape preserving representations and optimality of the Bernstein basis

This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bézier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ) is also totally positive.Both authors were partially supported by DGICYT PS90-0121.     

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure.     

Gevrey Smoothness of Families of Invariant Curves for Analytic Area Preserving Mappings

In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. We specifically obtain the Gevrey index of families of invariant curves which is related to the smoothness of area preserving mappings and the exponent of small divisors condition. Moreover, we obtain a Gevrey normal form of area preserving mappings in a neighborhood of the union of the invariant curves.     

Smoothness Properties of Modified Bernstein-Kantorovich Operators

In this article, we consider modified Bernstein-Kantorovich operators and investigate their approximation properties. We show that the order of approximation to a function by these operators is at least as good as that of ones classically used. We obtain a simultaneous approximation result for our operators. Also, we prove two direct approximation results via the usual second-order modulus of smoothness and the second-order Ditzian-Totik modulus of smoothness, respectively. Finally, some graphical illustrations are provided.     

Diameter Two Properties,Convexity and Smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties.We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space \({X}\): “For every slice \({S}\) of \({B_X}\) and every norm-one element \({x}\) in \({S}\), there is a point \({y \in S}\) in distance as close to 2 as we want.” Spaces with this property are shown to have non-smooth bidual.     

Shape restriction of the multi-dimensional Bernstein prior for density functions

Random Bernstein polynomials induces a probability measure on the space of multivariate density functions on a unit cube. For density estimation, it is important that the Bernstein prior can be restricted to an admissible class of densities with certain geometric properties of the target density. In this article, we study the shape properties such as monotonicity, convexity, and symmetry of the Bernstein prior.     

Meyer-KnigandZeller算子的保形问题

讨论了 Meyer-Knig and Zeller算子的保形逼近问题 ,我们用基于算子特殊结构的分析方法得到了该算子的保单调性 .保凸性以及保形逆定理等保形性质 .     

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator B _n fixing a strictly positive function f ₀, and a second function f ₁ such that f ₁/f ₀ is strictly increasing, within the framework of extended Chebyshev spaces U _n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B _n : C[a, b] → U _n with strictly increasing nodes, fixing f₀, f₁ ? U_n{f_{0}, f_{1} \in U_{n}} . If U_n ì U_{n + 1}{U_{n} \subset U_{n + 1}} and U _n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B _n+1 : C[a, b] → U _n+1 with strictly increasing nodes, fixing f ₀ and f ₁. In particular, if f ₀, f ₁, . . . , f _n is a basis of U _n such that the linear span of f ₀, . . . , f _k is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B _n with increasing nodes fixing f ₀ and f ₁. The second main result says that under the above assumptions the following inequalities hold

On Smoothness Properties and Approximability of Optimal Control Functions

The theory of discretization methods to control problems and their convergence under strong stable optimality conditions in recent years has been thoroughly investigated by several authors. A particularly interesting question is to ask for a natural smoothness category for the optimal controls as functions of time.In several papers, Hager and Dontchev considered Riemann integrable controls. This smoothness class is characterized by global, averaged criteria. In contrast, we consider strictly local properties of the solution function. As a first step, we introduce tools for the analysis of L elements at a point. Using afterwards Robinson's strong regularity theory, under appropriate first and second order optimality conditions we obtain structural as well as certain pseudo-Lipschitz properties with respect to the time variable for the control.Consequences for the behavior of discrete solution approximations are discussed in the concluding section with respect to L as well as L ₂ topologies.     

Properties of fractional calculus with respect to a function and Bernstein type polynomials

This paper is divided into two stages. In the first stage, we investigated a new approach for the $\psi$-Riemann–Liouville fractional integral and the Faa di Bruno formula for the $\psi$-Hilfer fractional derivative. In addition, we discussed other properties involving the $\psi$-Hilfer fractional derivative and the $\psi$-Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the $\psi (·)$ function are investigated and the $\psi$-Riemann–Liouville fractional integral and $\psi$-Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the $\psi$-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro-differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.     

延迟更新序列半群的某些性质

称一序列(v_n,n>1）是延迟更新序列当且仅当它是转移概率序列(p_(ij)（n）,n>1）。在本文我们证明了延迟更新序列半群具有性质ILID（即无穷小阵的极限是无穷可分的）,并且证明了正延迟更新序列半群是一Delphic半群。     

关于半群的一些性质定理的证明

本文从推导非双射变换可以作成一个半群出发,论证了任何半群和变换半群的关系,并推出了半群的广义结合律和广义交换律.     

Properties of Bernstein functions of several complex variables

A multidimensional generalization of the class of Bernstein functions is introduced and the properties of functions belonging to this class are studied. In particular, a new proof of the integral representation of Bernstein functions of several variables is given. Examples are considered.     

单纯形上Bernstein多项式的一些性质

设Ｂ_m（ｆ，·）为函数ｆ在ｄ维单纯形σ上的ｎ阶Ｂｅｒｎｓｔｅｉｎ多项式，本文对ｆ∈Ｃ^ｒ（σ）及ｆ∈Ｃ^r+2（σ）给出了ｆ的各阶编导数用Ｂ_n（ｆ，·）相应偏导数逼近的误差估计．同时也考虑了整系数Ｂｅｒｎｓｔｅｉｎ多项式的Ｌ_p模估计     

多元推广的Bernstein算子的逼近性质

构造了一类一致收敛于被逼近函数的多元序列,以此序列为基础,运用多元函数的全连续模及部分连续模来刻画这种多元推广的Bernstein算子的逼近性质,不仅得出了理论逼近结果,而且给出了数值逼近的例子.     

拟对角扩张Cuntz半群的某些性质

设O→J→A→B→O是一个拟对角扩张.作者证明如果J和B具有Cuntz半群的某些性质,则A也具有相同的半群性质.     

Shape Preserving Interpolation to Convex Data by Rational Functions with Quadratic Numerator and Linear Denominator

By using rational functions of the type quadratic/linear (withquadratic numerator and linear denominator), we show in thispaper how to construct, in a straightforward way, a convex C¹interpolant to convex data. In Delbourgo & Gregory (1985b)a cubic/quadratic rational form is discussed for this problem.A special case arises there which allows a reduction to thequadratic/linear form of the present paper. This simpler formwas not evident at the time and we give here an independentaccount of the relevant theory which we support by numericalexamples. Finally we examine the consistency conditions (a setof nonlinear equations) for second-derivative continuity. Weprove that a unique solution exists which satisfies the convexityrequirements.