Metric Projections in Spaces of Continuous Functions

Speaker: Aldric Loughman Brown

Affiliation: University College London

Time: Tuesday 14/06/2011 from 14:00 to 15:00

Venue: Access Grid UWS. Presented from Parramatta (EB.1.32), accessible from Campbelltown (26.1.50) and Penrith (Y239).

Abstract: Let T be a topological space (a compact subspace of Rm, say) and let C(T) be the space of real continuous functions on T, equipped with the uniform norm: ∥f∥ = maxt∈T |f(t)| for all f ∈ C(T). Let G be a finite dimensional linear subspace of C(T). If f ∈ C(T) then


d(f, G) = inf{∥f − g∥ : g ∈ G}

is the distance of f from G, and

PG(f) = {g ∈ G : ∥f − g∥ = d(f, G)}

is the set of best approximations to f from G. Then

PG :C(T)→P(G)

is the set-valued metric projection of C(T) onto G. In the 1850s P. L. Chebyshev considered T = [a, b] and G the space of polynomials ofdegree ≤ n − 1. Our concern is with possible properties of PG. The historical development, beginning with Chebyshev, Haar (1918) and Mairhuber (1956), and the present state of knowledge will be outlined. New results will demonstrate that the story is still incomplete.

Biography: Aldric was one of the numerous research students of
Dr Frank Smithies from Cambridge - with him, Aldric embarked on the process of becoming a Functional Analyst, but by the time he wrote a Ph.D. thesis he was an Approximation Theorist. A large part of his research has concerned a variety of problems of best approximation, often related to the subject of the talk Aldric gave in Ballarat on 25th May 2011. He is a member of the Editorial Board of the Journal of Approximation Theory. Aldric's own research students (two women, three men) have been one from each of Trinidad, India, Palestine, England and Iran.