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Charles Fefferman 30/01/2018, 17:00 — 18:00 — Abreu Faro Amphitheatre
, Princeton University

Interpolation and Approximation in Several Variables

Let $X$ be our favorite Banach space of continuous functions on $\mathbb{R}^n$ (e.g. $C^m$ , $C^{m,α}$ , $W^{m,p}$). Given a real-valued function $f$ defined on an (arbitrary) given set $E$ in $\mathbb{R}^n$ , we ask: How can we decide whether $f$ extends to a function $F$ in $X$? If such an $F$ exists, then how small can we take its norm? What can we say about the derivatives of $F$? Can we take $F$ to depend linearly on $f$?

What if the set $E$ is finite? Can we compute an $F$ whose norm in $X$ has the smallest possible order of magnitude? How many computer operations does it take? What if we ask only that $F$ agree approximately with $f$ on $E$? What if we are allowed to discard a few points of $E$ as “outliers”; which points should we discard?

A fundamental starting point for the above is the classical Whitney extension theorem.

The results are joint work with Arie Israel, Bo’az Klartag, Garving (Kevin) Luli, and Pavel Shvartsman.

See also


CAMGSD CEMAT IT Instituto Superior Técnico Fundação para a Ciência e a Tecnologia