De Branges Type Lemma and Approximation in Weighted Spaces

G. Paltineanu,I. Bucur

doi:10.1007/s00009-021-01764-y

Abstract

The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

Highlights

Using two fundamental tools in functional analysis: Hahn–Banach and Krein– Milman theorems, in 1959, Louis de Branges [2] give a nice proof of Stone– Weierstrass theorem on algebras of real continuous functions on a compact Hausdorff space
We present, in a natural way, the duality theory for general weighted spaces, i.e. a class of scalar continuous functions on a locally compact space (Lemma 2.1, Theorem 2.1)
We extend de Branges Lemma in this new setting: Lemma 3.2 for linear subspaces and Lemma 5.1 for convex cones of a weighted space

Summary

Introduction

Using two fundamental tools in functional analysis: Hahn–Banach and Krein– Milman theorems, in 1959, Louis de Branges [2] give a nice proof of Stone– Weierstrass theorem on algebras of real continuous functions on a compact Hausdorff space. We present, in a natural way, the duality theory for general weighted spaces, i.e. a class of scalar continuous functions on a locally compact space (Lemma 2.1, Theorem 2.1). Branges Lemma for weighted spaces was obtained by Prolla in [5], in the same particular case Beside these Lemmata, Theorems 3.1 and 5.1 play a crucial role in the proof of various approximations results: Theorems 4.1–4.5 as well as Corollaries 4.1–4.3. The locally convex topology defined by this family of seminorms is denoted by ωV and it will be called the weighted topology on CV0(X). We mention some particular weighted spaces: a) If V = {1} CV0(X) = C0(X)—the space of continuous functions vanishing at infinity and the weighted topology ωV coincide with the uniform convergence topology.

Duality for Weighted Spaces

Lemma De Branges for Weighted Spaces

Some Approximation Theorems in Weighted Spaces

Stone–Weierstrass Theorem for Convex Cones in Weighted Spaces