Properties of Finite DifferencesI have seen this statement in different variants, but could not find...
Properties of Finite Differences
I have seen this statement in different variants, but could not find a proof:
If the n-th order differences of equally spaced data are non-zero and constant then the data can be represented by a polynomial.
I interpret this statement more formally as follows:
Let be any function such that the finite differences , defined by
for (where by convention)
are such that is constant and non-zero for any . Then f is a polynomial in x (with Q coefficients) and degree .
Question: Is this correct as I stated it? How does one prove that such f must be a polynomial?
Answer & Explanation
Every polynomial of degree d can be expressed in the basis given by the Falling Factorials up to falling index d.
Then, from the properties of the falling factorial, it is easy to demonstrate that
And viceversa, given
the Newton series assures us that
So we can conclude that
if we are given a sequence of m+1 points equally spaced by h
and the corresponding sequence of values taken by a function f(x)
we can put
denoting the differences as
we have that the sequence in n of is the Binomial Transform of the sequence of the values g(k+n)
and since the Binomial Transform is self-inverse, we get
which is precisely the Newton series.
given n+1 points and the respective g(k+j)the Newton series is the unique polynomial of max degree n which interpolates them.
if the differences of order n<m are all equal to each other (at changing k)
then all the differences of order higher than n will be null
and the relevant Newton series will be the unique polynomial of degree n which interpolates all the m+1 points