Date & Time: Tuesday, October 14, 2014, 11:30-12:30.
Venue: Room 216

Title: Best Uniform Approximation to Continuous Functions from Finite Dimensional Spaces of Continuous Functions - from 1855 to 2013

Speaker: Aldric Brown, University College London

Abstract: In the 1850’s, P. L. Chebyshev considered best uniform approximations to a function $ f : [0, 1] \rightarrow \mathbb{R} $ from the space of real polynomial functions of degree less than or equal to $ n - 1 $. He discovered the phenomenon of uniqueness of best approximations and characterised them - Approximation Theory was born. Now one considers a more general situation. Let $ T $ be a compact Hausdorff space (for example an interval, a circle, a square); let $ C(T ) $ be the space of real continuous functions $ f : T \rightarrow \mathbb{R} $, equipped with the uniform norm $ || f || = \max \{ |f(t)| : t \in T \} $ ; let $ M $ be a finite dimensional subspace of $ C(T ) $. If $ f $ is in $ C(T ) $, then we define the distance \[ dist(f, M ) = \min \{ f - g : g \in M \} \] and we let \[ P_M(f) = \{ g \in M : f - g = dist (f, M) \] denote the set of best uniform approximations to $ f $ from $ M $. So $ P_M : C(T ) \righatarrow \{ W : W \subset M \} $ is a set-valued mapping; it is called the metric projection of $ C(T ) $ onto $ M $. For each $ f $ in $ C(T) $, $ P_M (f) $ is a non-empty compact convex subset of $ M $. In the 1950’s there arose (naturally, from a result in the theory of integral equations) the question whether, given $ M $ there exists a continuous selection for $ P_M $, that is, a continuous $ s : C(T ) \rightarrow M $ such that $ s(f) \in P_M (f) $ for all $ f $ in $ C(T) $. If $ P_M $ is lower semi-continuous then the answer is ‘Yes’ (Michael, 1956). Since then there has been a steady development of a theory of these metric projections. The talk will describe how recent results (2013) have changed the story, brought it closer to its origins, and generated hope for the unanswered questions.