Chebyshev Subspaces Research Articles

Chebyshev Subspaces of Dirichlet Series

Chebyshev Subspaces of Dirichlet Series

Best Co-Approximation In Weighted Space

In this paper, we present best co- approximation in weighted space. The results considered are these of existence of functions of best co- approximation, specifications of co-a proximal and specification co- Chebyshev subspaces.

Finite Dimensional Chebyshev Subspaces of Lo

If A is a subset of the normed linear space X, then A is said to be proximinal in X if for each xÎX there is a point y0ÎA such that the distance between x and A; d(x, A) = inf{||x-y||: yÎA}= ||x­-y0||. The element y0 is called a best approximation for x from A. If for each xÎX, the best approximation for x from A is unique then the subset A is called a Chebyshev subset of X. In this paper the author studies the existence of finite dimensional Chebyshev subspaces of Lo.

The linearity coefficient of metric projections onto one-dimensional Chebyshev subspaces of the space C

The paper deals with the operator of metric projection onto an arbitrary one-dimensional Chebyshev subspace 〈φ〉 of the space C[K] of real-valued functions defined and continuous on a Hausdorff compact set K. The linearity coefficient of the operator is calculated in terms of the parameters of the generating function φ. As a consequence, a new estimate of the Lipschitz constant of the operator is obtained.

2-Chebyshev subspaces in the spaces L 1 and C

The 2-uniqueness subspaces and the finite-dimensional 2-Chebyshev subspaces of the space C of functions continuous on a Hausdorff compact set and of the space L1 of functions Lebesgue integrable on a set of σ-finite measure are described. These descriptions are analogs of the well-known Haar and Phelps theorems for ordinary Chebyshev subspaces.

The unicity of best approximation in a space of compact operators

Chebyshev subspaces of $\mathcal{K}(c_0,c_0)$ are studied. A $k$-dimensional non-interpolating Chebyshev subspace is constructed. The unicity of best approximation in non-Chebyshev subspaces is considered.

The unicity of best approximation in a space of compact operators

Chebyshev subspaces of $ \mathcal{K}(c_0, c_0)$ are studied. A k-dimensional non-interpolating Chebyshev subspace is constructed. The unicity of best approximation in non-Chebyshev subspaces is considered.

The linearity of metric projection operator for subspaces of L p spaces

Let Y be a Chebyshev subspace of a Banach space X. Then the single-valued metric projection operator PY: X ∈ Y taking each x ∈ X to the nearest element y ∈ Y is well defined. Let M be an arbitrary set and µ be a σ-finite measure on some σ-algebra Σ of subsets of M. We give a description of Chebyshev subspaces Y ⊂ Lp(M,Σ, µ) with a finite dimension and a finite codimension which the operator PY is linear for.

Best approximation in Chebyshev subspaces of L(l1n,l1n)

Chebyshev subspaces of \(\mathcal{L}(l_1^n,l_1^n)\) are studied. A construction of a \(k\)-dimensional Chebyshev (not interpolating) subspace is given.

ON ω-CHEBYSHEV SUBSPACES IN BANACH SPACES

The purpose of this paper is to introduce and discuss the concept of <TEX>${\omega}$</TEX>-Chebyshev subspaces in Banach spaces. The concept of quasi Chebyshev in Banach space is defined. We show that <TEX>${\omega}$</TEX>-Chebyshevity of subspaces are a new class in approximation theory. In this paper, also we consider orthogonality in normed spaces.

Proximinality in Banach spaces

In this paper, we study approximatively τ-compact and τ-strongly Chebyshev sets, where τ is the norm or the weak topology. We show that the metric projection onto τ-strongly Chebyshev sets are norm- τ continuous. We characterize approximatively τ-compact and τ-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in τ-almost locally uniformly rotund spaces. We also prove some stability results on approximatively τ-compact and τ-strongly Chebyshev subspaces.

ε-Pseudo Chebyshev and ε-quasi Chebyshev subspaces of Banach spaces

We will define and characterize ε-pseudo Chebyshev and ε-quasi Chebyshev subspaces of Banach spaces. We will prove that a closed subspace W is ε-pseudo Chebyshev if and only if W is ε-quasi Chebyshev.

Approximating weak Chebyshev subspaces by Chebyshev subspaces

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties.

ε-weakly Chebyshev subspaces of banach spaces

We will define and characterize ε-weakly Chebyshev subspaces of Banach spaces. We will prove that all closed subspaces of a Banach space X are ε-weakly Chebyshev if and only if X is reflexive.

On copositive approximation in some classical spaces of sequences

In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces and shows that it is unique.

Alexandroff's Double Arrow Compact Space and Approximation Theory

A compact space Q similar to the compact space known as Alexandroff's double arrow space is constructed. It is shown that the real space C(Q) has no Chebyshev subspaces of codimension >1, but the complex space C(Q) has such subspaces.

On Weak Chebyshev Subspaces and the Property awc1 a Survey

The property awc1 was used by many authors to obtain results concerning approximation by elements of weak Chebyshev subspaces. In this paper the author studies this property in details, by collecting the scattered results concerning it and by finding new ones.

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 .

Linearity of metric projections on Chebyshev subspaces inL 1 andC

LetY be a Chebyshev subspace of a Banach spaceX. Then the single-valued metric projection operatorP Y :X → Y taking eachx ∈X to the nearest elementy ∈ Y is well defined. LetM be an arbitrary set, and letμ be aσ-finite measure on someσ-algebra gS of subsets ofM. We give a complete description of Chebyshev subspacesY ⊂L 1(M, Σ,μ) for which the operatorP Y is linear (for the spaceL 1[0, 1], this was done by Morris in 1980). We indicate a wide class of Chebyshev subspaces inL 1(M, Σ,μ), for which the operatorP Y is nonlinear in general. We also prove that the operatorP Y , whereY ⊂C[K] is a nontrivial Chebyshev subspace andK is a compactum, is linear if and only if the codimension ofY inC[K] is equal to 1.

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).