## Distributional Convergence of Hermite functions?

By Hermite functions, I mean Hermite polynomials, multiplied with the factor $e^{-x^2/2}$ and some normalizing constant.

Now the first Hermite function $\psi_0$ is basically the density of normal distribution and as known, if you define $\psi_0^{\varepsilon}(x) = 1/\varepsilon \cdot \psi_0(1/\varepsilon)$, then this family, if you think of it as family of distributions (in the dual of the Schwarz space), converges to the delta distribution.

My question is: If you define $\psi_n^\varepsilon$ in the same way as above, where $\psi_n$ is the $n$th hermite function, does this converge in the distributional sense, and to what?

My first hypothesis was that it converges to the $n$th derivative of $\delta$ and I thought it was a straightforward calculation, but apparently, it is not that easy.

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If you take any $f\in L^1(\mathbb{R})$, and define $f^\epsilon(x):=\frac{1}{\epsilon}f(x/\epsilon)$ then $f^\epsilon$ converges to the delta at zero times $\int_{\mathbb{R}}fdx$. – Pietro Majer Oct 10 2011 at 9:46