Let $X$ be your favorite Banach space of continuous functions on $\mathbb{R}^n$. Given a real-valued function $f$ defined on some (possibly awful) set $E$ in $\mathbb{R}^n$, 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? Can we make $F$ depend linearly on $f$? What can we say about the derivatives of $F$ at or near points of $E$ (assuming $X$ consists of differentiable functions)?

Suppose $E$ is finite. Can we compute a nearly optimal $F$? How many computer operations does it take? What if we demand merely that $F$ agree approximately with $f$? Suppose we allow ourselves to discard a few data points as “outliers”. Which points should we discard?

The video of this talk is available on the IISc Math Department channel.