# What are the best known bounds on the Hermite polynomials?

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

Has this been improved?

• What kind of bounds? pointwise? Lp norm? The polynomials themselves or the spectral coefficients related to them? – Amir Sagiv Apr 21 '16 at 22:23
• 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! – gradstudent Apr 21 '16 at 22:46
• I'd include that in the question. – 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.