I came across an interesting application of interpolation in the area of logic and a notion of stability in interpolation methods. Stability is defined as the resulting function converging to the approximated function regardless of the measurement points as long as their numbers converges to infinity. The polynomial interpolation or spline interpolations we learn in class do satisfy this criteria.

Fuzzy logic deals with how we reason with approximations rather than precise values such as those deduced from predicate logic. For instance, decisions about how cold it is cannot be made with absolute certainty and in applications that require such data, we use fuzzy algorithms to make certain decisions. The first step of every such algorithm requires a set of human defined rules that systems can convert to mathematical equivalents. For instance, we can define a person who is taller than some x meters to be considered TALL by the system. The important thing here is that the rule set has to be complete. The system would have to know a state for every possible value of x. Hence these algorithms often end up running into problems of storage space and exponential runtime. A possible solution was to use an incomplete set of rules whereby interpolation is used to determine the ‘fuzziness’ of the variables that weren’t defined. Thus we have a few interpolation methods proposed in this article which compares a few fuzzy logic interpolation techniques and their relative effectiveness.

http://www.mft.hu/hallg/200104.pdf
http://en.wikipedia.org/wiki/Fuzzy_logic