calculation – hermit polynomial $ H_ eta ( psi) $ where $ psi $ is a function that appears in quadrature of Gauss-Hermite

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 by

begin {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.

fa.functional analysis – On the functions obtained from the integration of Gauss's quadrature

Fix an integer $ n ge $ 2. Let $ x_1, …, x_n $ sand $ w_1, …, w_n $ s are the nodes of Gaussian quadrature and the weights respectively in the meantime $[0,1]$ (Https://en.wikipedia.org/wiki/Gaussian_quadrature). As in On the continuity and injectivity of the Gaussian quadrature scheme for digital integration, with a function of identical weight $ 1 $ , to define $ T_n: C ([0,1]) to C ([0,1]$ as $ T_n (f) (x) = x sum_ {i = 1} ^ n w_i f (xx_i), forall f in C ([0,1]), forall x in [0,1]$ (We use the formula as obtained in the answer in the link).

In the related question, it has been proved that each of these $ T_n $ is a linear continuous function on $ (C ([0,1]), ||. || _ infty) $ . And also that $ T_n $ converge is $ ||. || _ infty $ standard of the operator to $ T $, or $ T (f) (x) = int_0 ^ xf (t) dt $.

My questions are now:

(1) What is the closure of $ Im T_n $ ?

(2) leave $ Lip [0,1]$ refers to the set of all functions of Lipschitz on $[0,1]$. What is the closure of $ Lip[0,1] cap Im T_n $ ?

(3) What is the closure of $ C ^ 1[0,1] cap Im T_n $ ?