Fix an integer $ n \ge 2$ . Let $ x_1,…,x_n$ s and $ w_1,…,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $ [0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . As in On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $ 1$ , 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 are using the formula as obtained in the answer in the link).

In the linked question , it has been proven that each such $ T_n$ is a linear continuous function on $ (C([0,1]), ||.||_\infty)$ . And also that $ T_n$ converges is $ ||.||_\infty$ operator norm to $ T$ , where $ T(f)(x)=\int_0^x f(t) dt$ .

My questions now are the following :

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

(2) Let $ Lip [0,1]$ denote the set of all Lipschitz function 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$ ?