I'm struggling and would like some help understanding and beginning to prove the statement below. This is much different than anything we've done so far in class so any assistance is appreciated.
Prove any translation of the ReLu function $\rho (x-a)$ has a polynomial approximation, for $0\leq a \leq 1$. (A direct proof of this, via Taylor series, is outlined in the text.)
i.e. We want to prove that for any $\epsilon>0$ there exists a polynomial $p(x)$ such that $\displaystyle\sup_{∈[0,1]}|(−)−()|<$.
Some definitions
A function $\phi \colon [0,1] \to \mathbb{R}$ is said to be polygonal if it is continuous, and there is a partition of $[0,1]$ \begin{equation} 0= x_0 < x_1 < \cdots < x_n=1 \end{equation} so that $ \phi $ is linear on each interval $x _{k-1} < x < x _{k}$, for $k=1 ,2 , \ldots, n $.
and
The RELU function is $ \rho (x) = \max (0,x) $. It was used by Henri Lebesgue in 1898 to give a proof of the Weierstrass Approximation Theorem. The argument has three steps. Use/prove them to prove the Weierstrass Approximation Theorem.
The proof mentioned in the question showed that:
$\sqrt{1-x}=\displaystyle\sum_{n=0}^ \infty a_n x^n$
for $a_0=1$, and $a_n=\frac{-1\cdot3\cdot5\cdots(2n-3)}{2\cdot4\cdot6\cdots2n}$ for $n\geq1$
I believe the connection here is that $\frac{\sqrt{1+x}^2 -1+\sqrt{x^2}}{2}=\frac{x+\vert x\vert}{2}$.