In trying to find the error bound of polynomial interpolation for evenly spaced nodes I need to show $$|(x-x_1)(x-x_2)...(x-x_n)| \leq \frac{h^nn!}{4n}$$ I know that this maximum has to happen somewhere between the first two or last two nodes, but when I tried to reformulate the problem like this I just ended up shifting the function along the x axis. Trying to find a pattern in the expanded version of the polynomial did not help either and at this point im just completely at a loss for what to try.
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In the range $x_1\le x\le x_n$, $x$ must lie in some interval $x_i\le x\le x_{i+1}$. It is easy to see that $|(x-x_i)(x-x_{i+1})|\le\frac{h^2}{4}$ by considering the maximum/minimum of the quadratic on the range $[x_i,x_{i+1}]$.
Then taking the maximum values of $|x-x_j|$ for $j<i$ and $j>i+1$, $$|(x-x_1)\cdots(x-x_i)(x-x_{i+1})\cdots(x-x_n)|\le \frac{h^2}{4}\cdot 2h\cdots ih\cdot2h\cdots(n-i)h$$ Clearly, the worst cases are when $i=1$ or $n-1$. In either of these cases $$|(x-x_1)\cdots(x-x_i)(x-x_{i+1})\cdots(x-x_n)|\le \frac{h^2}{4}\cdot(2h)(3h)\cdots (n-1)h=\frac{h^n(n-1)!}{4}$$
Chrystomath
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