Elements Of Best Approximation Research Articles

Approximation of Non-Analytic Functions by Analytical Ones

We study the properties of the elements of best approximation for functions defined in the unit disk by functions from the Bergman space. For functions of a special type, we find a sufficiently accurate description of the properties of these elements in terms of the Hardy and Lipschitz classes. The obtained result is based on an analysis of the corresponding duality relation for extremal problems. The developed method is also applicable to relatively smooth (in terms of Sobolev spaces) approximated functions.

Best Approximation in Metric Spaces

AbstractThe aim of this paper is to prove some results on the existence and uniqueness of elements of best approximation and continuity of the metric projection in metric spaces. For a subset M of a metric space (X; d), the nature of set of those points of X which have at most one best approximation in M has been discussed. Some equivalent conditions under which an M-space is strictly convex have also been given in this paper.

Best Approximation By Upward Sets

In this paper we prove some results on upward subsets of a Banach lattice �� with a strong unit. Also we study the best approximation in �� by elements of upward sets, and we give the necessary and sufficient conditions for any element of best approximation, by a closed subset of �� .

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.

On the Problem of Distance From a Point to the Subspace of Finite Codimension in Real Banach Space

In this paper,assuming X is Banach and L is a finite codimension subspace of X. Application of the dual method,there obtain the formula of distance and elements of best approximation from given a point to the subspace of finite codimension under some condition,and apply the formula to solve some of the specific optimal control problem.

On uniform constants of strong uniqueness in Chebyshev approximations and fundamental results of N. G. Chebotarev

In the problem of the best uniform approximation of a continuous real-valued function in a finite-dimensional Chebyshev subspace , where is a compactum, one studies the positivity of the uniform strong uniqueness constant . Here stands for the strong uniqueness constant of an element of best approximation of , that is, the largest constant such that the strong uniqueness inequality holds for any . We obtain a characterization of the subsets for which there is a neighbourhood of satisfying the condition . The pioneering results of N. G. Chebotarev were published in 1943 and concerned the sharpness of the minimum in minimax problems and the strong uniqueness of algebraic polynomials of best approximation. They seem to have been neglected by the specialists, and we discuss them in detail.

Computability of finite-dimensional linear subspaces and best approximation

We discuss computability properties of the set P G ( x ) of elements of best approximation of some point x ∈ X by elements of G ⊆ X in computable Banach spaces X . It turns out that for a general closed set G , given by its distance function, we can only obtain negative information about P G ( x ) as a closed set. In the case that G is finite-dimensional, one can compute negative information on P G ( x ) as a compact set. This implies that one can compute the point in P G ( x ) whenever it is uniquely determined. This is also possible for a wider class of subsets G , given that one imposes additionally convexity properties on the space. If the Banach space X is computably uniformly convex and G is convex, then one can compute the uniquely determined point in P G ( x ) . We also discuss representations of finite-dimensional subspaces of Banach spaces and we show that a basis representation contains the same information as the representation via distance functions enriched by the dimension. Finally, we study computability properties of the dimension and the codimension map and we show that for finite-dimensional spaces X the dimension is computable, given the distance function of the subspace.

Computability of the Metric Projection Onto Finite-dimensional Linear Subspaces

We show that given a computable Banach space X and a finite-dimensional subspace U of X the set of elements of best approximation of x ∈X (by elements of U) can be computed as a compact set with negative information. If X is uniformly convex, we can even compute the (unique) element of best approximation. Furthermore, given a uniformly convex computable Banach space X the mapping U ↦PU that maps each finite dimensional linear subspace to the corresponding (single-valued) metric projection is computable.

Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W, x), where x e X and W is a closed downward subset of X.

Extension of bounded linear functionals and best approximation in spaces with asymmetric norm

The present paper is concerned with the characterization of the elements of best approximation in a subspace \(Y\) of a space with asymmetric norm, in terms of some linear functionals vanishing on \(Y\). The approach is based on some extension results, proved in Section 3, for bounded linear functionals on such spaces. Also, the well known formula for the distance to a hyperplane in a normed space is extended to the nonsymmetric case.

Best approximation by polynomials

In this paper we show that if E is a separable Banach space, F is a reflexive Banach space, and n, k ∈ ℕ, then every continuous polynomial of degree n from E into F has at least one element of best approximation in the Banach subspace of all continuous k–homogeneous polynomials from E into F.

Characterization of best approximations in metric linear spaces

Let (X,d) be a real metric linear space, with translation-invariant metric d and G a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X.We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.

On a Discrete Kolmogorov Criterion

In this note we consider best approximation in 1-norm from subspaces of real n -dimensional spaces. Some discrete Kolmogorov-type criteria characterizing elements of best approximation are verified.

On meromorphic approximation

Let G be a bounded N-connected domain, the boundary Γ of which consists of closed analytic Jordan curves. We assume that 0eG. For any nonnegative integers n and m, denote by ℳ n,m the class of all meromorphic functions on G that can be represented in the form h=p/qz m , where p belongs to the Smirnov class E ∞(G), q is a polynomial of degree at most n, q≢0. Theorems giving necessary and sufficient conditions for a function belonging to the class ℳ n,m to be an element of best approximation to a continuous function f on Γ in the space L ∞(Γ) by functions in the class ℳ n,m are proved. Some questions concerning orthogonal polynomials and the theory of Hankel operators are also considered.

Haar problem for sign-sensitive approximations

The Haar problem for sign-sensitive approximations consists in finding necessary and sufficient conditions for a finite-dimensional subspace of the space of continuous functions on a compact subset of and a sign-sensitive weight , , ensuring that for each function in there exists a unique element of best approximation with weight . Several conditions of this kind are established. These conditions are shown to be closely connected with the topological properties of the annihilators of the functions and . In particular, the sign-sensitive weights are described such that the same condition as the one introduced by Haar for uniform approximations (that is, for ) serves the corresponding Haar problem.

A Problem in Linear Matrix Approximation

AbstractLet A be a normal operator in ℬ︁(H), H a complex Hilbert space, and let ℛ A = ≷ {AX ‐ XA:X ∈ ℬ︁(H)} be the commutator subspace of ℬ︁(H) associated with A. If B in ℬ︁(H) commutes with A, then B is orthogonal to ℛA with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ℛ A. This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p‐norm recently.We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in RA and prove that the metric projection of H onto ℛA is continuous.

ON SOME APPROXIMATION PROBLEMS IN METRIC SPACES


 
 
 In this paper we consider the problem of simultaneous characterization of a set of elements of best approximation and characterization of elements of $\varepsilon$-approximation in metric spaces. 
 
 

ON FRÉCHET SPACES WITH CERTAIN CLASSES OF PROXIMAL SUBSPACES

A new metric with absolutely convex balls is introduced on a metrizable locally convex space. Necessary and sufficient conditions are given for all closed hypersubspaces and all nonnormable closed subspaces of a Frechet space to be proximal, i.e., to have the property that there exist elements of best approximation with respect to this metric. In particular, these conditions are expressed in terms of the topologies of the original space and the strong dual space. It is proved that the Frechet spaces B × ω have the proximality property, where B is a reflexive Banach space and ω = RN is the nuclear Frechet space of all numerical sequences. Questions of Albinus and Wriedt are answered. Bibliography: 23 titles.

The existence of elements of best approximation in the complex space C(Q)

The existence of elements of best approximation in the complex space C(Q)

Subspaces of finite codimension: The existence of elements of best approximation

Subspaces of finite codimension: The existence of elements of best approximation