Weak Chebyshev subspaces and alternation

Frank Richard Deutsch, Günther Nürnberger and Ivan Singer

Vol. 89 (1980), No. 1, 9–31

DOI: 10.2140/pjm.1980.89.9

Abstract

Let T be a locally compact subset of R and C₀(T) the space of continuous function which vanish at infinity. An n dimensional subspace G of C₀(T) may possess one of the three alternation properties:

1. For each f ∈ C₀(T) which has a unique best approximation g₀ ∈ G, f −g₀ has n + 1 alternating peak points;
2. For each f ∈ C₀(T), there exists a best approximation g₀ ∈ G to f such that f − g₀ has n + 1 alternating peak points;
3. For each f ∈ C₀(T) and each best approximation g₀ ∈ G to f,f − g₀ has n + 1 alternating peak points.

In this paper, for each i ∈{1,2,3} we give an intrinsic characterization of those subspaces G of C₀(t) which have property (A-i).

Mathematical Subject Classification 2000

Primary: 41A50

Milestones

Received: 2 August 1978

Revised: 12 July 1979

Published: 1 July 1980