How to compute orthonormal polynomials from weight function?

Question

I have a weight function $w(x)=e^{-x}$ with $x$ from $0$ (inclusive) to infinity. How would I compute the first four orthonormal polynomials with respect to this weight function?

Answer 1

You need to use Gram-Schnidt process to the set of polynomials

$$\left\{ 1,x,x^2, x^3\right\} $$

using the inner product

$$ <f,g> = \int_{0}^{\infty} f(x)g(x) e^{-x}dx . $$

Here is the first two $\left\{ 1,(x-1)\right\}$.

Answer 2

According to the table of orthogonal polynomials, these are Laguerre Polynomials: they are solution to Laguerre's Differential Equation and can be also found using $L_n(x)=\frac{1}{n!} \left( \frac{d}{dx} -1 \right) ^n x^n$. Here are the first four: $\left\{1,\; -x+1,\; \frac{1}{2}(x^2-4x+2),\; \frac{1}{6}(-x^3+9x^2-18x+6)\right\}$.

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