A Schauder basis of Schwartz space.

Question

It is well-known that hermite functions $\{h_n(x)\}_n$ form a Schauder basis of the Schwartz space $\mathcal{S}(\mathbb{R})$. Let $\alpha, \beta \in \mathbb{R}^*$. Does the 'modified' family of functions $\{\alpha h_n(\beta x)\}_n$ still form a Schauder basis of $\mathcal{S}(\mathbb{R})$ ??

Answer 1

COMMUNITY WIKI: VAGUE ANSWER, USE AS A DRAFT

I vaguely remember a "dilation formula" or "linearization formula", something like that, unfortunately I forgot the exact nomenclature. It reads $$ H_n(\beta x) =\sum_{j=0}^n a_{j, n, \beta} H_j(x),$$ for $\beta >0$. This is relevant here.

Indeed, you can forget about that $\alpha$, it is irrelevant, if a set is Schauder then $\alpha$ times that set is still Schauder. The problem is that $\beta$. The formula I vaguely remember might take care of that.