The Hermite functions $(h_n)_{n \ge 0}$ are the eigenfunctions of $T f = (x+\partial_x)(x-\partial_x) f$, they are defined by their generating function $g(x,t)=\sum_{n=0}^\infty h_n(x) t^n = e^{-x^2/2+2xt-t^2}$, looking at $\int_{-\infty}^\infty g(x,t)g(x,u)dx= \sqrt{\pi} e^{2tu}$, $\|h_n\|_{L^2}^2 = \sqrt{\pi} \frac{2^n}{n!}$, $\partial_x g,\partial_t g$, $(x-\partial_x) h_n = (n+1)h_{n+1},(x+\partial_x) h_n = 2 h_{n-1}$ shows that
$\left(\frac{h_n}{\|h_n\|_{L^2}}\right)_{n \ge 0}$ is an orthonormal basis of the Schwartz space, in the sense that it is an orthonormal basis of $L^2(\Bbb{R})$ and $f \in \mathscr{S}(\Bbb{R})$ if and only if $f = \sum_{n \ge 0} c_n \frac{h_n}{\|h_n\|_{L^2}}$ where $\forall k, \lim_{n \to \infty} c_n n^k=0$.
A similar behavior appears with $(e^{2i \pi nx})_{n \in \Bbb{Z}}$ the eigenfunctions of $f \mapsto {}{}{}{}{}{} i\partial_x f$, they are an orthonormal basis of $C^\infty(\Bbb{R/Z})$ again in the sense that they are an orthonormal basis of $L^2(\Bbb{R/Z})$ and $f \in C^\infty(\Bbb{R/Z})$ iff $f = \sum_n c_n e^{2i \pi nx}$ where $\forall k, \lim_{n \to \pm\infty} c_n n^k=0$.
Question : What would be some other relevant examples of such basis ? For what kind of $\scriptstyle\text{(differential, normal, compact resolvent)}$ operator can we hope its eigenfunctions will be such an orthonormal basis of the Schwartz space ?