# On functions obtained from Gaussian Quadrature integration

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