## 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?

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 $$C ^ 1[0,1] cap Im T_n$$ ?