If you have observed outcomes 
at time or location 
, how can you represent
the general relationship between time and observation? There are several
interpolation results - classes of functions to `fit' observed data. One
key observation is
Weierstrass Approximation Theorem Let f be a continuous function on
[a,b]. Then given any 
, there exists a polynomial P on [a,b] such that
.
This suggests that polynomial approximation is a useful avenue for study.
Consider, for example, fitting a polynomial through the points
(0,10), (1,9), (2,0). Because there are three points, a quadratic is the
appropriate polynomial to use. Let 
. We have
. Solving, we find 
.
It is interesting to note that, even though these three points are monotonically
decreasing, the polynomial fit is not - P(0.5)=10.5, bigger than the observed
largest value.
More generally, simple fitting requires solution of a linear system for the coefficients:
A different approach with similar ends is to generate the Lagrange interpolation
polynomial. For simplicity, consider a linear function at first,
, which is supposed to go through
points 
. Written as above, the 0-subscripted part of the
polynomial should vanish at 
, and vise versa. Plugging in, we find
More generally, the interpolation polynomial 
is built from the factors
as
EXERCISE: Compute the Lagrange polynomial through the three points used above.