The Interpolation Problem in 1D

I am actually interested in the interpolation problem in high dimension;
however, that problem turns out to be surprisingly difficult to approach.
Therefore, in this talk, I want to warm up for that problem by going over
some natural interpolation approaches in the lowest possible dimension,
where we can actually draw pictures of what we're doing.

The interpolation problem asks us to connect the dots between a
list of n pairs of data points (x,y).  We might think there's
only one way to do this, but there are actually a variety of techniques,
each making certain choices about the allowable class of answers
before trying to select the best one.

There are many things we could ask from an interpolant, including
continuity, efficiency, monotonicity, convexity, and locality.  We'll 
look at some common 1D interpolants, consider their advantanges, and
ask whether they can be efficiently extended to higher dimensional
problems.

The familiar cases of nearest neighbor, piecewise linear, and polynomial
interpolants will be augmented by a discussion of Shepard and Radial
Basis Function interpolants.