**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.