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


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