Let $$f_n(x) = \frac{d^4}{dx^4}P_n(x)$$
where $P_n(x)$ are the Legendre polynomials. By which polynomial should one multiply this family to construct a family or orthogonal polynomials on $[-1,1]$ with weight factor $1$?
I have no idea where to start with this. Looking for some hints how to even begin. What theory is relevant here?
Best Answer
You have to multiply $f_n$ by $(1-x^2)^2$ to get an orthogonal family of polynomials. In general, if $\{P_n\}_{n\ge 1}$ is the system of orthogonal polynomials with respect to the weight $W(x)$ then $k-$ the derivatives form an orthogonal system with respect to the weight $W^k(x)a(x)$ where $b(x)$ comes from the differential equation for orthogonal polynomials: $$a(x)y''(x)+b(x)y'+\lambda_n y=0.$$
In this particular case, Legandre polynomials arise as solutions of the differential equation $$(1-x^2)y''(x)-2xy'+n(n+1)y=0.$$
and $W=1$ which makes $4- $th derivatives to be orthogonal with respect to the weight $(1-x^2)^4.$ In other words $(1-x^2)^2f_n$ are going to be orthogonal with respect to the weight $1.$
One way to prove the statement above is to use the given differential equation together with integration by parts.