Best uniform line for a fractional power
= Best uniform line for a fractional power
{title2=$p^*(x)=x+\tfrac12(1-k)k^{k/(1-k)}$}
For $0<k<1$, $x^k-x$ is nonnegative on $[0,1]$ and has its maximum $(1-k)k^{k/(1-k)}$ at $x=k^{1/(1-k)}$. Shifting the chord $x$ upwards by half this maximum creates three alternating errors of equal magnitude. The <equioscillation theorem> proves that the displayed line is the unique best uniform linear approximation, with error equal to its constant shift.