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.     

联合逼近的存在性与离散化

本文研究了紧距离空间上的函数集合被满足Young条件的函数类的联合逼近问题,建立了存在性定理,得到了极限定理,证明了可以用函数集合的离散化或距离空间的离散化得到最佳联合逼近。     

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.     

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.     

Some new results on approximation in fuzzy 2-normed spaces

In this paper, we define a fuzzy 2-normed space and study the concept of best approximation in fuzzy 2-normed spaces (FTNS). We also define a set of best approximations, proximal set and approximatively compact set and prove some interesting results in this new setup.     

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．     

External Ellipsoidal Estimation of the Union of Two Concentric Ellipsoids and Its Applications

For an arbitrary set representable as the convex hull formed by the union of two concentric ellipsoids we propose a method to construct a family of external undominated ellipsoidal approximations and represent the estimated set as the intersection of all estimates from a given family. A sufficient condition of undominated guaranteed ellipsoidal approximation of a convex compactum is derived. A method is described that for certain classes of sets (such as the intersection of an ellipsoid or a cone with two halfspaces) constructs a family of internal undominated ellipsoidal approximations using the previous formulas for the external estimates of the union of concentric ellipsoids.     

Algorithms for the Best Simultaneous Uniform Approximation of a Family of Functions Continuous on a Compact Set by a Chebyshev Subspace

We generalize the cutting-plane method and the Remez method to the case of the problem of the best simultaneous uniform approximation of a family of functions continuous on a compact set.     

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.     

The Degree Of Polynomial Approximation And Interpolation Of Analytic Functions

The rate of best polynomial approximation of an analytic function on a compact Faber set K is characterized in terms of the rate of growth of its Faber coefficients and compared with the rate of approximation by the partial sums of the Faber series. Also the convergence of sequences of interpolating polynomials constructed for various systems of nodes is studied by considering the growth of the interpolated function. Under appropriate assumptions on K the approximation by interpolating polynomials can be incorporated in the characterization theorem. Emphasis is laid on high precision in describing the rate of approximation and on admitting a large class of functions.     

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.     

Best uniform approximation in the metric space of continuous mappings with compact convex images

For the problem of the best uniform approximation of a continuous mapping with compact convex images by sets of other continuous mappings with compact convex images, we establish necessary and sufficient conditions and a criterion for an element to be extremal; the criterion obtained is a generalization of the classic Kolmogorov criterion for a polynomial of the best approximation.     

Best approximation by linear combinations of characteristic functions of half-spaces

It is shown that for any positive integer n and any function f in with p∈[1,∞) there exist n half-spaces such that f has a best approximation by a linear combination of their characteristic functions. Further, any sequence of linear combinations of n half-space characteristic functions converging in distance to the best approximation distance has a subsequence converging to a best approximation, i.e., the set of such n-fold linear combinations is an approximatively compact set.     

τ-CHEBYSHEV AND τ-COCHEBYSHEV SUBPSACES OF BANACH SPACES

The concepts of quasi-Chebyshev and weakly-Chebyshev and σ-Chebyshev were defined [3 - 7], and as a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi[1]. In this research, we shall define τ-Chebyshev subspaces and τ-cochebyshev subspaces of a Banach space, in which the property τ is compact or weakly-compact, respectively. A set of necessary and sufficient theorems under which a subspace is τ-Chebyshev is defined.     

SimultaneousL 1 approximation of a compact set of real-valued functions

Summary In this paper the uniqueness results found in simultaneous Chebychev approximation are extended to simultaneousL ₁ approximation. In particular a sufficient condition to guarantee uniqueness of a best approximate to aL ₁ compact set is given.This paper is taken in part from a thesis to be submitted by M. P. Carroll in partial fulfillment of the requirements for the Ph. D. degree in the Department of Mathematics at Rensselaer Polytechnic Institute.     

τ-CHEBYSHEV AND τ-COCHEBYSHEV SUBPSACES OF BANACH SPACES

The concepts of quasi-Chebyshev and weakly-Chebyshev and σ-Chebyshev were defined [3 - 7], andas a counterpart to best approximation in normed linear spaces, best coapprozimation was introduced by Franchetti and Furi^[1]. In this research, we shall define τ-Chebyshev subspaces and τ-cochebyshev subspaces of a Banach space, in which the property τ is compact or weakly-compact, respectively. A set of necessary and sufficient theorems under which a subspace is τ-Chebyshev is defined.     

Multi-Input Multi-Output Ellipsoidal State Bounding

Ellipsoidal state outer bounding has been considered in the literature since the late sixties. As in the Kalman filtering, two basic steps are alternated: a prediction phase, based on the approximation of the sum of ellipsoids, and a correction phase, involving the approximation of the intersection of ellipsoids. The present paper considers the general case where K ellipsoids are involved at each step. Two measures of the size of an ellipsoid are employed to characterize uncertainty, namely, its volume and the sum of the squares of its semiaxes. In the case of multi-input multi-output state bounding, the algorithms presented lead to less pessimistic ellipsoids than the usual approaches incorporating ellipsoids one by one.     

Bounding a domain containing all compact invariant sets of the permanent-magnet motor system

In this paper we characterize a locus of compact invariant sets of the system describing dynamics of the permanent-magnet synchronous motor (PMSM). We establish that all compact invariant sets of this system are contained in the intersection of one-parameter set of ellipsoids and compute its parameters. In addition, localizations by using a parabolic cylinder, an elliptic paraboloid and a hyperbolic cylinder are obtained. Simple polytopic bounds are derived with help of these localizations. Most of localizations mentioned above remain valid for more specific motor systems; namely, for the interior magnet PMSM and for the surface magnet PMSM. Yet another localization set for the interior magnet PMSM is described. Examples of localization of chaotic attractors existing for some values of parameters of PMSMs are presented as well.     

The continuity of the metric projection on a subspace of finite codimension in the space of continuous functions

The closed subspaces of finite codimension of the space C(X) of all continuous real-valued functions on a compact Hausdorff space X, for which the set of elements of best approximations of every function f C(X) is nonempty and compact, are characterized. It is shown that if the compact Hausdorff space X is infinite, then C(X) has no subspace of a finite Codimension n > 1 which has a nonempty set of elements of the best approximation for an arbitrary function f 6 (X) and which has an upper-semicontinuous metric projection.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 531–539, April, 1976.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.