The best I could find on the net is this paper, http://arxiv.org/pdf/math/0401310.pdf

Has this been improved?

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    $\begingroup$ What kind of bounds? pointwise? Lp norm? The polynomials themselves or the spectral coefficients related to them? $\endgroup$ – Amir Sagiv Apr 21 '16 at 22:23
  • $\begingroup$ I would most importantly like to have bounds on sums like $\sum_{i=D}^\infty a^iH_i(x) H_i(y)$ where $a<1$ and $x, y \in \mathbb{R}$ and $D$ is some positive integer. Any bound you can help me find which will help bound sums like the above will be helpful! $\endgroup$ – gradstudent Apr 21 '16 at 22:46
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    $\begingroup$ I'd include that in the question. $\endgroup$ – Amir Sagiv Apr 22 '16 at 6:03

Not a comprehensive answer, but this paper contains some pointwise bounds on Hermite polynomials. It seems that it gives a different flavour of the $L^{\infty}$ bound by splitting it to $\mathbb{R}$ bounds and "central" bounds.

Also, for what it's worth, it seems that no one who's citing the aforementioned paper has a better bound.

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