On best uniform restricted range approximation in complex-valued continuous function spaces

We investigate the problem of best restricted range approximation of complex-valued continuous functions for a very general system of restrictions. Our results, including the characterizations, uniqueness and strong uniqueness, extend all recent results due to Smirnovs.     

Limit theory of restricted range approximations of complex-valued continuous functions

This paper is concerned with the problem of the best restricted range approximations of complex-valued continuous functions.Several properties for the approximating set(?)such that the classical characterization results and/or the uniqueness results of the best approximations hold are introduced.Under the very mild conditions,we prove that these properties are equivalent that(?)is a Haar subspce.     

On best simultaneous approximation in quotient spaces

We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.     

Best restricted range approximation

A theory of best restricted range approximation is developed for an extended n-dimensional Chebyshev subspace of C[a,b] of order n without restricting the upper and lower restraining functions. This theory includes a “zero in the convex hull” characterization, an alternation theorem, and a strong uniqueness theorem.     

On statistical approximation in spaces of continuous functions

Our goal is to present approximation theorems for sequences of positive linear operators defined on C(X), where X is a compact metric space. Instead of the uniform convergence we use the statistical convergence. Examples and special cases are also provided.      

Discretization and best approximation in Besov spaces

In this paper we prove a theorem about existence of best approximation in a class of spaces involving Besov spaces, via a discretization technique. It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B _∞,q ^a (π). It is also proved the proximinality of R _m ⁿ [a, b] in B _p,q ^a (π) for arbitrary p,q and a.     

Discretization and best approximation in Besov spaces

In this paper we prove a theorem about existence of best approximation in a class of spaces involving Besov spaces, via a discretization technique. It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B _∞,q ^a (π). It is also proved the proximinality of R _m ⁿ [a, b] in B _p,q ^a (π) for arbitrary p,q and a.     

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.     

On tractability of approximation in special function spaces

We consider approximation problems for a special space of d$d$ variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show that, depending on the norm for measuring the error, the problems are strongly polynomially or quasi-polynomially tractable even in the model of computation where functional evaluations have the cost exponential in the number of active variables.     

On the range of use of the tight-binding approximation method for a complex-valued potential

We consider a one-dimensional Schrödinger operator with a periodic potential constructed as the sum of the shifts of a given complex-valued potential q that decreases at infinity. Mathematical justification of the tight-binding approximation method is presented. Let λ₀ be an isolated eigenvalue of the Schrödinger operator with a potential q. Then, for the corresponding operator with a periodic potential, a continuous spectrum exists in the neighborhood of λ₀. The asymptotic behavior of this part of the spectrum as the period increases infinitely is studied for the cases of one- and two-dimensional invariant subspaces corresponding to λ₀.     

On isomorphisms of continuous function spaces

The following theorem, which strengthens the classical theorem of Stone, is proved: If there is an isomorphism φ ofC(X) ontoC(Y) (X, Y-compact) with ∥ φ ∥ · ∥ φ^?1 ∥ < 2, thenX andY are homeomorphic.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

Nonlinear weighted best simultaneous approximation in Banach spaces

The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S- and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets and strict RS-sets. Our results obtained in the present paper extend and improve all earlier known results in this direction.     

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.