In his 1972 PhD thesis "Simulation and Model Building for Mineral Processing" Whiten provides a visual comparison of the result of simulating a data set with various functions.
Similarly in SPOC (Simulated Processing of Ore and Coal, CANMET, circa 1985) visual and statistical comparisons are provided for simulation results from the same raw data for a variety of breakage and selection functions for ball miling.
Whiten shows that higher order polynomials can have some nasty end behaviours (flicking up like Dilbert's tie) and that misuse of polynomials can be troublesome. We can all see that palying around in Excel. It is always amusing to see a "fit" that is statistically the best (lowest sum of errors squared over all the data points) but a terrible fit in the area of intended use.
SPOC (Spring, Larsen, Mular) show that the selection functions in particular can be completely different around that critical size region depending on the function used.
Whiten developed a Fortan program (and provided the code listing) for fitting data to a multi-dimensional cubic spline as part of his thesis.
I like the cubic spline. Of course I like the APL implentation by Krol, June 81 most - because it can be used directly in METSIM. But a polynomial can be perfectly satisfactory in many cases, and using METSIM's matrix interpolation you can use a polynomial to generate data and then replace "nasty" values at the ends with nice smooth controlled curves. And if the polynomial is only used in a fixed range (such as in the component thermo data where the temperature range for use is limited and specified) then there is nothing wrong witha polynomial.
So we should be focused on results, goodness of fit for our use, and always aware of the limitations of our raw data before getting too worried about " a method".