Approximation of continuous set-valued maps by constant set-valued maps with image balls

It is shown that the problem of the best uniform approximation in the Hausdorff metric of a continuous set-valued map with finite-dimensional compact convex images by constant set-valued maps whose images are balls in some norm can be reduced to a visual geometric problem. The latter consists in constructing a spherical layer of minimal thickness which contains the complement of a compact convex set to a larger compact convex set.     

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.     

On polyhedral approximations in an <Emphasis Type="Italic">n</Emphasis>-dimensional space

The polyhedral approximation of a positively homogeneous (and, in general, nonconvex) function on a unit sphere is investigated. Such a function is presupporting (i.e., its convex hull is the supporting function) for a convex compact subset of Rⁿ. The considered polyhedral approximation of this function provides a polyhedral approximation of this convex compact set. The best possible estimate for the error of the considered approximation is obtained in terms of the modulus of uniform continuous subdifferentiability in the class of a priori grids of given step in the Hausdorff metric.     

Best Simultaneous Approximation and Its Applications in C ℝ(Q)

We develop a theory of best simultaneous approximation for closed convex sets in C _?(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max _q?Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C _?(Q) and give various characterizations of best simultaneous approximation.     

BEST MONOTONE L_■-APPROXIMATIONS IN SEVERAL VARIABLES

Given a continuous function f defined on the unit cube of R~n and a convexfunction _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set ofbest L~(t)-approximations by monotone functions has exactly one elementft,which is also a continuous function.Moreover if the family of convexfunctions {_t}t>0 converges uniformly on compact sets to a function _0,then the best approximation f_t→f_0 uniformly,as t→0,where fo is thebest approximation of f within the Orlicz space L~(0) The best approxima-tions{f_t}are obtained as well as minimizing integrals or the Luxemburgnorm     

Multi-segmental representations and approximation of set-valued functions with 1D images

In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as images are considered. For such a continuous SFV of bounded variation (CBV multifunction), we show that the boundaries of its graph are continuous, and inherit the continuity properties of the SVF. Based on these results we introduce a special class of representations of CBV multifunctions with a finite number of ‘holes’ in their graphs. Each such representation is a finite union of SVFs with compact convex images having boundaries with continuity properties as those of the represented SVF. With the help of these representations, positive linear operators are adapted to SVFs. For specific positive approximation operators error estimates are obtained in terms of the continuity properties of the approximated multifunction.     

KY FAN TYPE BEST APPROXIMATION THEOREM FOR A CLASS OF FACTORIZABLE MULTIFUNCTIONS

An existence result on Ky Fan type best approximation is proved. For this purpose, a class of factorizable multifunctions and the other one being a demicontinuous, relative almost quasi-convex, onto function on an approximately weakly compact, convex subset of Hausdorff locally convex topological vector space are used. As consequence, this result extends the best approximation results of Basha and Veeramani [8] and many others.     

On well posedness of best simultaneous approximation problems in Banach spaces

The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.     

Best Uniform Approximation of a Family of Functions Continuous on a Compact Set

We investigate the problem of the best uniform approximation of a function continuous on a compact set. We generalize the principal results of this investigation to the problem of the best simultaneous uniform approximation of a family of functions continuous on a compact set.     

On the strong CE-property of convex sets

We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered.     

Geometry and Topology of Continuous Best and Near Best Approximations

The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints.     

近严格凸与最佳逼近

本文研究近严格凸与最佳逼近的关系．证明了Ｂａｎａｃｈ空间Ｘ是近严格凸的当且仅当Ｘ的每个子空间是紧－半－切比晓夫空间．     

单隐层神经网络与最佳多项式逼近

研究单隐层神经网络逼近问题．以最佳多项式逼近为度量,用构造性方法估计单隐层神经网络逼近连续函数的速度．所获结果表明:对定义在紧集上的任何连续函数,均可以构造一个单隐层神经网络逼近该函数,并且其逼近速度不超过该函数的最佳多项式逼近的二倍．     

Failure of Polynomial Approximation on Polynomially Convex Subsets of the Sphere

An example is presented of a compact polynomially convex subsetof the unit sphere in C³ on which the polynomials in the complexcoordinate functions are not dense in the continuous functions.Also presented is an example of a compact polynomially convexsmooth solid torus lying in the unit sphere in C⁵ with the samefailure of polynomial approximation.     

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.     

Some best approximation results in locally convex spaces

In this note we obtain generalization of well known results of carbone and Conti, Sehgal and Singh and Tanimoto concerning the existence of best approximation and simultaneous best approximation of continuous funcitons from the set up of a normed space to the case of a Hausdorff locally convex space.     

On the convexity of images of nonlinear integral operators

We study continuous nonlinear Urysohn-type integral operators acting from the spaces of vector functions with integrable components to the space of continuous functions. We obtain conditions under which the images of sets defined by pointwise constraints have a convex closure under the action of these operators. The result is used to justify a method of constructive approximation of these images and to derive a necessary solvability condition for Urysohn-type integral equations. A numerical method for finding the residual of equations of this type on the sets under consideration is justified.     

Convergence analysis of an optimally accurate frozen multi-level projected steepest descent iteration for solving inverse problems

In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable.     

Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is analyzed. When applied to convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. In previous studies, for the approximation of a multidimensional ball, the convergence rates of the method were estimated in terms of the number of faces of all dimensions and the cardinality of the facial structure (the norm of the f-vector) of the constructed polytope was shown to have an optimal rate of growth. In this paper, the asymptotic convergence rate of the method with respect to faces of all dimensions is compared with the convergence rate of best approximation polytopes. Explicit expressions are obtained for the asymptotic efficiency, including the case of low dimensions. Theoretical estimates are compared with numerical results.