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 $ ?