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