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.