We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$. a) Show that if $f$ is $\delta$-Lipschitz, then $\left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { \delta } { 2 \sqrt { n } }$ for all integer $n \geqslant 1$. b) Deduce that if $f$ is of class $C ^ { 1 }$, then there exists a real $c$ such that, for all $n \in \mathbb { N } ^ { * } , \left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { c } { \sqrt { n } }$. c) Extend the previous result to the case where $f$ is a continuous function, piecewise $C ^ { 1 }$.
We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by
$$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$
for all $x \in [ 0,1 ]$.
a) Show that if $f$ is $\delta$-Lipschitz, then $\left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { \delta } { 2 \sqrt { n } }$ for all integer $n \geqslant 1$.\\
b) Deduce that if $f$ is of class $C ^ { 1 }$, then there exists a real $c$ such that, for all $n \in \mathbb { N } ^ { * } , \left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { c } { \sqrt { n } }$.\\
c) Extend the previous result to the case where $f$ is a continuous function, piecewise $C ^ { 1 }$.