We consider three specific kinds of problems:
1. Suppose we know the values of f(x) on two intervals, [a,b] and [c,d] but not in between [b,c].
we seek a function g(x) defined on [b,c] that "smoothly interpolates" between our two fragments of f.
2. We know f(x) at k specific points, x1 ,...,xk. That is, we know f(x1 ),...,f(xk ).
We want a smooth curve that links them.
3. As in 2. we want a polynomial of degree j with j < k - 1 that "best" fits our points.
We will present solutions to these. There are many variations, (suppose for example the values we have are uncertain) that are also useful.
Our general approach is based on the following claim:
A polynomial of lowest degree that meets all our conditions provides a reasonable smooth interpolation.