Approximation In Normed Linear Spaces Research Articles

Extensions of closed convex processes

In this paper we provide some norm preserving extension results for real valued closed convex processes. As a consequence, we characterize afterwards, the elements of best approximation in normed linear spaces by elements of closed convex cones using closed convex processes.

Generalized Contraction and Invariant Approximation Results on Nonconvex Subsets of Normed Spaces

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalizedF-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.

SOme results about best coapproximation in L P (S,X)

As a counterpart to best approximation in normed linear spaces, the best coapproximation was introduced by Franchetti and Furi. In this paper, we shall consider the relation between coproximinality M in X and L p(S,M) in L p(S,X). Finally we give some results in cochebyshev subspaces and additional subspaces.

New Characterization for Nonlinear Weighted Best Simultaneous Approximation

This paper is concerned with the problem of a wide class of weighted best simultaneous approximation in normed linear spaces, and it establishes a new characterization result for the class of approximation by virtue of the notion of simultaneous regular point.

Some Results on Best Coapproximation in L 1 (S, X)

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we shall show that if M is a separable, coproximinal subspace of X satisfying some additional conditions, then L1 (S, M) is coproximinal in L1(S, X).

Best coapproximation in quotient spaces

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we apply the above coapproximation, and obtain some results on the upper semi-continuity of cometric projection maps. Also we shall determine under what conditions coproximinality can be transmitted to and from quotient spaces.

τ-Chebyshev and τ-cochebyshev subpsaces of Banach spaces-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.

Approximation in normed linear spaces

A historical account is given of the development of methods for solving approximation problems set in normed linear spaces. Approximation of both real functions and real data is considered, with particular reference to L p (or l p ) and Chebyshev norms. As well as coverage of methods for the usual linear problems, an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters, and special attention is paid to some particular nonlinear approximating families.

Duality in quasi-convex supremization and reverse convex infimization via abstract convex analysis,and applications to approximation **

We give duality theorems and dual characterizations of optimal solutions for abstract quasi-convex supremization problems and infimization problems with abstract reverse convex constraint sets. Our main tools are dualities between families of subsets, conjugations of type Lau associated to them, and subdifferentials with respect to conjugations of type Lau. These tools permit us to give explicitly the relation between.the constraint sets, and the relation between the objective functions, of the primal problem and the dual problem. As applications, we obtain duality theorems for quasi-convex supremization and reverse convex infimization in locally convex spaces and, in particular, for worst and best approximation in normed linear spaces.

Approximations with a sign-sensitive weight: existence and uniqueness theorems

Approximations with a sign-sensitive weight generally take into account both the absolute value of the error of approximation and its sign. We study the problems of existence, uniqueness and plurality for the element of best uniform approximation with a given sign-sensitive weight by functions of a given family on an interval . We also study these problems for approximations in normed linear spaces by elements of a family , where the deviation of an element from another element is measured by the value of some non-negative sublinear functional . A very important role is played by the rigidity and freedom of the systems and . These notions are also studied in the paper, with special attention being given to the case of Chebyshev subspaces .

Best Interpolatory Approximation in Normed Linear Spaces

A theory of best approximation with interpolatory contraints from a finite-dimensional subspaceMof a normed linear spaceXis developed. In particular, to eachx∈X, best approximations are sought from a subsetM(x) ofMwhichdependson the elementxbeing approximated. It is shown that this “parametric approximation” problem can be essentially reduced to the “usual” one involving a certainfixedsubspaceM0ofM. More detailed results can be obtained when (1)Xis a Hilbert space, or (2)Mis an “interpolating subspace” ofX(in the sense of [1]).

A generalization of theorems of Eidelheit and Carleman concerning approximation and interpolation

We prove necessary and sufficient conditions for linear operators to approximate and interpolate unbounded continuous functions on certain subsets U ⊆ (−∞, ∞). The main application of our general theory is to simultaneous asymptotic approximation and interpolation by function series. Special cases of our results are a sharpened version of a theorem of Eidelheit for the solubility of infinite systems of linear equations and a generalization of a theorem of Carleman concerning the asymptotic approximation and interpolation of continuous functions by entire functions on the real axis. Moreover we can apply our general theorems to a moment problem of Pólya and to asymptotic approximation and interpolation by Dirichlet series. Our general approach to such problems is based on the use of certain complete approximation systems and on an essential identity theorem of functional analysis concerning approximations in normed linear spaces with certain additional restrictions by seminorms.

Approximation by linear combinations of continuous functions with restricted coefficients

We prove necessary and sufficient conditions for linear combinations of given functions K k ϵ C[ a, b] ( k = 1, 2, …) to be dense in C[ a, b], C 0[ a, b] respectively, where the coefficients satisfy bound constraints. Our general approach to such problems is based on a new method of functional analysis concerning a relationship between approximations in normed linear spaces with generalized restrictions, defined by seminorms, and certain corresponding properties of bounded linear functionals on these spaces. In the special case of approximation by Müntz polynomials with restricted coefficients some known results and sharpened versions of these can be deduced from our general theorems. Finally, an application to another type of function K k is obtained, where K k ( t) = φ( tλ k −1), λ k → ∞, with φ being certain analytic functions.

Uniform non-linear constrained approximation in normed linear spaces

The theory of H-sets as originally defined by L. Collatz has been developed in recent papers and shown to be a unifying concept for linear approximation problems. We here extend the theory of H-sets for non-linear uniform approximation with functional constraints where the approximating functions have a compact domain and range in a Banach space. With the theory of H-sets as here developed the characterization of global and local best approximations follow, and conditions for local to be global approximations are given. A convergence theorem for a descent algorithm is given.

Characterization of best approximations in normed linear spaces of matrices by elements of finite-dimensional linear subspaces

A general characterization theorem of best approximations in normed linear spaces is specialized to the linear space of real n × n matrices endowed with the spectral norm.

Uniform linear approximation in normed linear spaces

It has been shown that the theory of H-sets is important in the characterization of best uniform approximation of continuous real-or complex-valued functions. We here extend the theory of H-sets to the more general setting of functions with compact domain and with range contained in a Banach space. Using the definitions of H-sets, we construct a maximal linear functional and obtain inclusion theorems analogous to the classical case. It is then a simple matter to deduce a characterization of best approximation and show when uniqueness and strong uniqueness are achieved.

On best simultaneous approximation in normed linear spaces

In this paper we consider the problem of simultaneous approximation of a subset F of a Banach space B by elements of another subset S ⊂ B. Results are obtained on the existence, uniqueness, and characterization of best simultaneous approximations.

Perturbations of best approximation problems

The purpose of this paper is to unify a number of continuity, perturbation, and discretization properties of best approximations in normed linear spaces or in metric or semimetric spaces. It extends work of Aubin and, more specifically, of Daniel.

Book Review: Best approximation in normed linear spaces by elements of linear subspaces

Book Review: Best approximation in normed linear spaces by elements of linear subspaces

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory