Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.     

The similarity between the best approximation and the best copositive approximation

In this paper the author studies the copositive approximation in C(?) by elements of finite dimensional Chebyshev subspaces in the general case when ? is any totally ordered compact space. He studies the similarity between me behavior of the ordinary best approximation and the behavior pf the copositive best approximation. At the end of this paper, the author isolates many cases at which the two behaviors are the same.     

On the best approximation of vector valued functions by polynomials with coefficients in vector spaces

In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain direct and inverse theorems for the best approximation by generalized polynomials and results concerning the existence (and uniqueness) of best approximation generalized polynomials. This paper was written during the 2005 Spring Semester when the second author (S.G. Gal) was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, USA. Mathematics Subject Classification (2000) 41A65, 41A17, 41A27     

Common fixed-points for Banach operator pairs in best approximation

The notion of Banach operator pairs is introduced, as a new class of noncommuting maps. Some common fixed-point theorems for Banach operator pairs and the existence of the common fixed-points of best approximation are presented. These results are proved without the assumption of linearity or affinity for either f or g, which shows that the concept about Banach operator pairs is potentially useful in the study of common fixed-points.     

The equivalence relations between the best approximation and linear operators approximation in a Besov space

In this paper, we characterize Besov type space introduced by the best approximation, best approximating elements and a kind of linear operators. The project is supported by NSFC     

On computing best Chebyshev complex rational approximants

An algorithm for computing best complex ordinary rational functions is presented. The final step of the procedure consists of solving the system of nonlinear equations defined by the local Kolmogorov criterion before checking recently developed sufficient optimality and uniqueness conditions. Various numerical results are reported exhibiting, in particular, nonunique solutions, saddle points and locally best approximants that are not global.     

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

Stability of the linear inequality method for rational Chebyshev approximation

The linear inequality method is an algorithm for discrete Chebyshev approximation by generalized rationals. Stability of the method with respect to uniform convergence is studied. Analytically, the method appears superior to all others in reliability.     

Adaptive bivariate Chebyshev approximation

We propose an adaptive algorithm which extends Chebyshev series approximation to bivariate functions, on domains which are smooth transformations of a square. The method is tested on functions with different degrees of regularity and on domains with various geometries. We show also an application to the fast evaluation of linear and nonlinear bivariate integral transforms. Work supported by the research project CPDA028291 “Efficient approximation methods for nonlocal discrete transforms” of the University of Padova, and by the GNCS-INdAM.     

A new Chebyshev polynomial approximation for solving delay differential equations

The purpose of this study is to give a Chebyshev polynomial approximation for the solution of mth-order linear delay differential equations with variable coefficients under the mixed conditions. For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated Chebyshev expansion of the function in the delay differential equations. Hence, the resulting matrix equation can be solved, and the unknown Chebyshev coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

Finite-codimensional Chebyshev subspaces in the complex spaceC(Q)

We consider finite-condimensional Chebyshev subspaces in the complex spaceC(Q), whereQ is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real spaceC(Q) contains finite-codimensional Chebyshev subspaces, then the same is true of the complex spaceC(Q) (with the sameQ). Translated fromMatermaticheskie Zametki, Vol. 62, No. 2, pp. 178–191, August, 1997. Translated by V. E. Nazaikinskii     

Best and coupled best approximation theorems in abstract convex metric spaces

In this paper, we first give a best approximation theorem in abstract convex metric spaces. As applications, we then derive some best and coupled best approximations and coupled coincidence point results in normed spaces and hyperconvex metric spaces.     

Best uniform approximation to a class of rational functions

We explicitly determine the best uniform polynomial approximation to a class of rational functions of the form 1/²(x−c)+K(a,b,c,n)/(x−c) on [a,b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n−1 denotes the degree of the best approximating polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle η in the representation of the approximation error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best approximating polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform approximation to some more functions.     

Isotone cones in Banach spaces and applications to best approximations of operators without continuity conditions

In this paper, we introduce the concept of isotone cones in Banach spaces. Then, we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity conditions in partially ordered Banach spaces.     

OnM-ideals and best approximation

In this paper we will prove some theorems on theM-ideals of compact operators and the best approximation of quasitriangular operator algebras. These results improve and extend the known results in [4, 5, 7].This work is supported in part by the National Natural Science Foundation of China.     

Gateaux differentiability in Orlicz–Lorentz spaces and applications

Let Λ_w,? be the Orlicz–Lorentz space. We study Gateaux differentiability of the functional ψ_w,? (f) = ? (f *)w and of the Luxemburg norm. More precisely, we obtain the one‐sided Gateaux derivatives in both cases and we characterize those points where the Gateaux derivative of the norm exists. We give a characterization of best ψ_w,? ‐approximants from convex closed subsets and we establish a relation between best ψ_w,? ‐approximants and best approximants from a convex set. A characterization of best constant ψ_w,? ‐approximants and the algorithm to construct the best constant for maximum and minimum ψ_w,? ‐pproximants are given. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Perturbation analysis for best approximation and the polar factor by subunitary matrices

This paper is a continuation and improvement over the results of Laszkiewicz and Zietak [BIT, 2006, 46: 345–366], studying perturbation analysis for polar decomposition. Some basic properties of best approximation subunitary matrices are investigated in detail. The perturbation bounds of the polar factor are also derived.