Orthonormal basis of the Hilbert space $L^2(\left[0,+\infty\right[, dx)$?

Question

I am searching for an orthonormal basis of the Hilbert space $L^2(\left[0,+\infty\right[, dx)$.

Thank you in advance

Answer 1

Laguerre polynomials $\{L_n\}$ are an orthogonal basis of the weighted space $L^2([0,\infty),e^{-x}\,dx)$. Then $\{e^{-x/2}L_n\}$ is an orthogonal basis of $L^2([0,\infty),dx)$.

Answer 2

The change of variable $t=e^{-s}$ gives $$ \int_{0}^{1}f(t)g(t)dt = \int_{0}^{\infty}f(e^{-s})g(e^{-s})e^{-s}ds $$ This may be interpreted as a unitary map $U : L^2[0,1)\rightarrow L^2[0,\infty)$ defined by $$ (Uf)(s) = e^{-s/2}f(e^{-s}). $$ The transform is isometric, meaning that $$ \|f\|_{L^2[0,1)} = \|Uf\|_{L^2[0,\infty)}. $$ And the change of variable is invertible, which proves that $U$ is unitary. Therefore, the unitary map $U$ maps orthonormal bases of $L^2[0,1)$ to orthonormal bases of $L^2[0,\infty)$. In particular, $$ \{ e^{-s/2}\exp(2\pi ine^{-s}) \}_{n=-\infty}^{\infty} $$ is a complete orthonorml basis of $L^2[0,\infty)$.