Past exam of the approximation theory and finite element methods course of the University of Oxford / 2009 / A / 3 / a

By definition, a best uniform approximation $p^{\ast }\in P_n$ minimizes $\Vert u-p{\Vert }_{\infty }$ over $p\in P_n$. Write $E=\Vert u-p^{\ast }{\Vert }_{\infty }$. The equioscillation theorem says that, if $E>0$, optimality is equivalent to the existence of $n+2$ ordered points with signed errorsThe absolute-value bars on the left of the printed formula must be omitted for this alternating-sign statement. A nonnegative absolute value cannot equal a negative $-E$.

For sufficiency, suppose a polynomial $p$ had strictly smaller error. At the alternating points, $p-p^{\ast }$ must have the same sign as $u-p^{\ast }$, since otherwise the new error would have magnitude at least $E$. The difference polynomial therefore changes sign in all $n+1$ successive gaps and has at least $n+1$ distinct roots, impossible for a nonzero element of $P_n$.

For necessity, suppose that no $n+2$ alternating extremal points exist. On the compact extremal set $K=\{x:|u(x)-p^{\ast }(x)|=E\}$, group the signs of the error into consecutive blocks. The positive and negative extremal sets are disjoint compact sets, so sign changes are separated by positive distances; the number of blocks is finite. The absence of an $n+2$ alternating selection means there are at most $n+1$ blocks. Place a root in each gap between successive blocks and choose the overall sign of their product $q$ to agree with the error on $K$. Then $\mathrm{deg}q\le n$ and $q$ is nonzero throughout $K$.

By compactness and continuity, on a neighborhood of $K$ the signs still agree and $|q|$ is bounded below. A sufficiently small positive $\eta$ reduces the error there when $p^{\ast }$ is replaced by $p^{\ast }+\eta q$. On the compact complement of this neighborhood the old error is below $E$ by a fixed positive margin, and choosing $\eta$ smaller preserves that margin. This gives a strictly better approximation, a contradiction. If $E=0$, the target already lies in $P_n$ and optimality is immediate.

For $u(x)=x^{1/3}$, the chord is $x$. Concavity gives $x^{1/3}-x\ge 0$, with maximum at $x_{\ast }=1/(3\sqrt{3})$ and maximum value $2/(3\sqrt{3})$. Shift the chord up by half that amount. The errors at $0,x_{\ast },1$ are respectively $-E,+E,-E$, so