• Speaker: Dr Riki Brown, University College London and University of Newcastle Upon Tyne
  • Title: Metric Projections in Spaces of Continuous Functions
  • Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Time and Date: 3:00 pm, Wed, 15th Jun 2011
  • Abstract:

    Let $T$ be a topological space (a compact subspace of ${\mathbb R^m}$, say) and let $C(T)$ be the space of real continuous functions on $T$, equipped with the uniform norm: $||f|| = \text{max}_{t\in T}|f(t)|$ for all $f \in C(T)$. Let $G$ be a finite-dimensional linear subspace of $C(T)$. If $f \in C(T)$ then $$d(f,G) = \text{inf}\{||f−g|| : g \in G\}$$ is the distance of $f$ from $G$, and $$P_G(f) = \{g \in G : ||f−g|| = d(f,G)\}$$ is the set of best approximations to $f$ from $G$. Then $$P_G : C(T) \rightarrow P(G)$$ 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 of degree $\leq n − 1$. Our concern is with possible properties of $P_G$. 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.

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