Let $P(x)=\sum_{i=0}^p a_i x^i$ be a polynomial of degree $p$ such that $\sum_{i=1}^p |a_i| \leq 1$ and $x \in [0,1]$.
Let $L_n(x)$ denote a piece-wise linear function on $[0,1]$. In other words, for every $L_n(\cdot)$, there exists a partition of unit interval $0=x_0 < x_1 <x_2<\ldots <x_{n-1}<x_n=1$ such that $L_n(x)$ is affine whenever $x_{i-1} \leq x \leq x_i$.
I want to find minimum number of partitions $N^\ast(p,\varepsilon)$ needed for piece-wise approximation of $P(\cdot)$ upto an error of approximation $\varepsilon$ in sup-norm. Equivalently, find minimum $n$ such that one can always find a $L_n(\cdot)$ satisfying $$ \|P(\cdot)-L_n(\cdot)\|_{\infty} \leq \varepsilon. $$
Can anyone guide me about how to solve this or point me to relevant references if this has been already solved before? My attempt to solve this problem has been to use linear interpolation between the points $0,\frac{1}{n},\frac{2}{n},\ldots,\frac{n-1}{n},1$ and use that to approximate error. But I am not sure if this is the right way to solve the problem since $L_n(\cdot)$ can have discontinuities.