For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{1}$. Show that there exists a constant $M \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M 2^{-j}$. Deduce that the sequence of functions $S_{n} f$ is uniformly convergent on $[0,1]$ when $n$ tends to $\infty$.
For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by
$$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$
where, for all $(j, k) \in \mathcal{I}$,
$$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$
Suppose $f$ is of class $\mathcal{C}^{1}$. Show that there exists a constant $M \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M 2^{-j}$.
Deduce that the sequence of functions $S_{n} f$ is uniformly convergent on $[0,1]$ when $n$ tends to $\infty$.