Integration with unbounded spectral weight

Question

I want to consider the solution of an equation in the space $L^2(e^{-2\alpha x} \mathbb{R}. )$ I am just wondering how to implement the weight when computing the norm.

So far, I use Hermite spectral method and I work out my weights by quadrature Q. I compute norm of a variable, q,in $L^2 (\mathbb{R}) $ space by simply doing $q^TQq$, and I’ve tried adjusting this by letting $q’ = e{-\alpha x$’ and then doing the quadrature but so far nothing is really working...

Kindest regards,

Catherine

Answer 1

With a change of variable $y = 2 \alpha x$, and depending upon the details of what you’re doing, the Gauss-Laguerre quadrature rules (see for example Stroud and Secrest) might prove fruitful.