I am working with a meteorologist on a project. We are pulling data from METAR Observation stations on several variables (such as temperature, dew point, wind speed, etc.) throughout time. Unfortunately there are occasional missing values for some of these variables at different times, but our model requires there to be no missing data at all.
I am from a mathematics background, so I was investigating some of the "pure math" ways to come up with a replacement value where data is missing. I have looked at Linear Interpolation, Cubic Spline Interpolation, Akima Spline Interpolation, and Polynomial Interpolation.
Other than the polynomial interpolation (because of the oscillatory problems) I have found in my simulations that all of the three methods predict the missing values very well, especially when data points are close together in time. However they all fall short at extreme values. I really thought that since I have past and future values to work with (not just past values), I should be able to find a method of replacement that can give better results at the maximum and minimum values. I have tried different averaging techniques that use values on both sides, and more.
Do you know of a method of interpolation (or maybe regression) that performs well at extreme values?
EDIT: I think that I did not explain my problem well. I know about Runge's phenomenon and that is why I tried three different spline methods, but I am looking for interpolation methods to help approximate the extreme values of the function $f$, not the extreme left and right values of $x$.