We are given that a set of polynomials on [-1,1] have the following properties and have to show they are unique by induction. I have a way to show they are unique, but is not what he is looking for. I honestly have never seen it presented this way.
P_n(x) = Ʃa_in*x^i
All P_n are mutually orthogonal
P_n(1) = 1
I know they are Legendre polynomials and I've derived them using Gram-Schmidt, but not sure what to try here.
We have to find P_0 and P_1 which are 0 and x respectively and then assume there are some P_n and P_n* that both satisfy the definition above and we show they are identical.
The Attempt at a Solution
His only hint was that through showing P_n is unique, you would need P_n-1 and should use induction. All I've ever done is take a difference, you get a polynomial of lower degree and after a bit of algebra, you get that the difference in the two must be zero so they are identical. That does only apply to monic polynomials, but it is easy to redefine P_n so that is monic.