When reading a technical paper (1) using Gauss-Hermite's quadrature, the following statements appear and I do not understand what it means. $ H_ eta ( psi) $ or $ psi $ is a function as below. Could someone explain the meaning?

Gauss-Hermite's quadrature is a modification of Gauss's quadrature

approaching the integration of a function between the -1 and

+1 as a weighted sum of function values at specified points in the integration domain. It introduces a function in decomposition

$ e ^ {- psi ^ 2} $, to extend the limits to $ – infty $ and $ + infty $:begin {equation} int _ {- infty} ^ { infty} e ^ {- psi ^ 2} f ( psi) d psi approx sum_ {m = 1} ^ anda w_m f ( psi_m) end {equation} Right here $ eta $ is

the number of samples according to the level set by the user

approximation while psi_m refers to the roots of the Hermite

polynomial, $ H_ eta ( psi) $. The corresponding weights for each sample

the points are given bybegin {equation} w_m = frac {2 ^ { eta – 1} eta! sqrt { pi}} { eta ^ 2 (H _ { eta-1} ( psi_m)) ^ 2}. end {equation}

I looked for it on Google but I could not find an explanation on this type of formulation, even if I found a lot on the case where $ H_ eta (x_i) $ or $ x_i $ is just a sampling point.

(1) O'Callaghan, Simon T. and Fabio T. Ramos. "Occupation maps of Gaussian processes." The International Journal of Robotics Research 31.1 (2012): 42-62.