João B. Prolla

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlins extension to normed spaces of Kakutanis Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.

Best Simultaneous Approximation

Publisher Summary This chapter considers (F, e· e) as a nontrivial nonarchimedean valued division ring and (E, ee· ee) is a nonzero nonarchimedean normed space over (F, e· e). The chapter discusses a problem that in the nonarchimedean case has a very simple solution: when M ≠ {O}, there is no uniqueness. A closed linear subspace M ⊂ E is called proximinal if P M (x) contains at least one element for all x ∈ E. This definition poses two problems—namely, (1) let M ⊂ E be a closed linear subspace. Give sufficient conditions on M so that M is proximinal—that is, every x ∈ E has at least one best approximation in M and (2) give sufficient conditions on E so that every closed linear subspace is proximinal. A continuous linear map P: E →E such that P 2 = P is called a continuous linear projection from E onto P(E).

An extension of Nachbin’s theorem to differentiable functions on Banach spaces with the approximation property

(1) for every xE U, there exis ts fEA such t h a t f ( x ) r (2) for every pair x, yEU, with xr there exists fEA such that f(x)r (3) for every xE U and vEE, with v ~ 0 , there exists fEA such that Df(x)v~O. In [1], Lesmes gave sufficient conditions for a subalgebra AcCm(E) to be dense in (Cm(E), z~), when m = l , and E is a real separable Hilbert space. In fact, he proved that (1), (2), (3) (with U=E) and

Weierstrass-Stone theorems for set-valued mappings

Let X be a locally compact Hausdorff space and (E, I( . 11) a normed space over IK (= IR or C). If W is a vector subspace of ~O(X; E), the space of all continuous functions f: X-+ E that vanish at infinity, and u, is a mapping from X into the non-empty subsets of E, we are interested in finding necessary and sufficient conditions under which, for every E > 0, there is some g E W such that g(x) E p(x) + {t E E; I/t (1 0, there is an e-approximate W-selection for ~0. More generally, we shall be interested in establishing a “localization formula” for the distance of a, from W:

The Approximation Property for Nachbin Spaces

Publisher Summary This chapter describes the approximation property for Nachbin spaces. In the chapter, X is a Hausdorff space that separates the points of X , and E is a non-zero locally convex space. The chapter proves that certain function spaces L ⊂ C (X;E) have the approximation property as soon as E has the approximation property. The chapter proves this for the class of all Nachbin spaces. Such spaces include C (X;E) with the compact-open topology; C b (X;E) with the strict topology; and C O (X;E) with the uniform topology. The techniques used in the chapter was suggested by Gierz, where he proved an analogue of a theorem for the case of X compact and bundles of Banach spaces. This technique of “localization” of the approximation property was used in the case of the partition by antisymmetric sets.

Simultaneous approximation and interpolation in weighted spaces

In this paper we present a result about simultaneous approximation and interpolation in weighted spaces. It generalizes a result of Prolla in the space of continuous functionsC(X;E) whereX is a compact Hausdorff space andE is a normed space. As a consequence, we prove that simultaneous approximation and interpolation is possible from certain vector subspaces.

Uniform Approximation of Continuous Functions with Values in [0,1]

Let X be a compact Hausdorff space and let D(X) denote the set of all continuous functions f from X into [0,1]. A subset A C D(X) has property V, by dennition, 1 — φ and φψ belong to A, whenever φ, ψ ∈ A. We prove a theorem of the Stone-Weierstrass type describing the uniform closure of A. Our result generalizes a theorem of R. I. Jewett that had provided a proof for a statement of von Neumann.

The Uniform Closure of Convex Semi-Lattices

Let X be a compact Hausdorff space, and let C(X IR) be the Banach lattice of all continuous real-valued functions on X with the sup-norm and pointwise ordering. We describe the uniform closure of convex (resp. convex conic) semi-lattices, i.e. inflattices and sup-lattices. Our result is an improvement on the classical Choquet-Deny Theorem.

On von neumann's variation of the weierstrass-stone theorem

Let X be a compact HausdorfF space and let D(X) be the set of all continuous real-valued functions f defined on X and such that 0 ≤ f(x) ≤ 1, for all x ∊ X. The set D(X) is equipped with the uniform topology. We characterize the uniform closure of subsets A ⊂ D(X) containing 0 and 1 and ϕψ + (1 − ϕ)η, whenever they contain ϕ, ψ and η

APPROXIMATION BY POSITIVE ELEMENTS OF SUBALGEBRAS OF REAL-VALUED FUNCTIONS

Abstract Let X be a compact Hausdorff space and let A be a subalgebra of C(X;IR) equipped with the sup-norm. Let A+ = {f ∈ A : f ≥ 0}. Then any positive f ∈ C(X; IR) belongs to the closure of A+ if and only if it verifies the following two conditions: (1) for any pair of points x, y such that f(x) ≠ f(y) there is some g ∈ A+ such that g(x) ≠ g(y); (2) for any point x such that f(x) > 0, there is some g ∈ A+ such that g(x) > 0.