When using polynomials to do spline interpolation or least-squares regression on a set of points which are not evenly spaced, there is a tendency for the polynomial to deviate wildly in those regions with a low point-density. I'm thinking about something like this:
Unfortunately for me, this intervening region is precisely the region I would like to interpolate, and the surrounding regions with a denser spacing of points are useful only insofar as they provide curvature information about this intervening region.
One workaround I've found is to simply place more points along the line connecting the boundary points of this region, however this is somewhat of a hack.
The cause of these wild deviations is of course the inflexibiltiy of the curvature (second derivative) of polynomials, so maybe I need to use another type of function? What are the standard methods for correcting for these deviations?