Hermite functions

Remarks

Example: $\psi_0(x) = \pi^{-1/4} e^{-x^2/2}$, and $\int_{\R} \psi_0(x)^2 \, dx = \pi^{-1/2} \int_{\R} e^{-x^2} \, dx = 1$. The orthogonality of the Hermite polynomials for the weight $e^{-x^2}$ gives $\int_{\R} \psi_m \psi_n \, dx = \delta_{mn}$. The family $(\psi_n)_{n \ge 0}$ is complete, so it is an orthonormal basis of $L^2(\R)$. Each $\psi_n$ solves $-\psi_n'' + x^2 \psi_n = (2n + 1) \psi_n$. With the Fourier transform $\hat{f}(\xi) = (2\pi)^{-1/2} \int_{\R} f(x) e^{-i x \xi} \, dx$, one has $\hat{\psi}_n = (-i)^n \psi_n$. Non-example: the polynomial $H_0 = 1$ without the Gaussian factor is not square integrable on $\R$, so it is not an element of $L^2(\R)$ at all.