Best uniform approximation

Given a family $A$ of approximating functions, a best uniform approximation to $f$ is an element $p\in A$ satisfying

Cosine substitution is an isometry from continuous functions on $[-1,1]$ to even continuous $2\pi$-periodic functions. It satisfies $\omega (\tilde{f},\delta )\le \omega (f,\delta )$ because cosine is Lipschitz continuous with constant one. Even degree-at-most-$n$ trigonometric polynomials correspond exactly to algebraic polynomials via $T_j(\cos \theta )=\cos (j\theta )$. Averaging a periodic approximant with its reflection cannot increase its error against an even target, so the algebraic and periodic best errors are equal.

If a bounded linear map $P$ reproduces every element of an approximating polynomial space $V$, then $f-Pf=(f-v)-P(f-v)$ for every $v\in V$. The triangle inequality and infimum over $v$ prove the displayed error bound. The same argument works for any linear approximating space, and applies in particular to de la Vallée Poussin sums.

For a fixed degree bound, two best approximating polynomials have a best approximating average. At each active point of that average their errors agree, hence the two polynomials agree. There must be at least two more active points than the degree: otherwise polynomial interpolation constructs a strictly improving sign-matching perturbation. The difference has too many zeros and must vanish. Finite-dimensional coefficient compactness also gives existence.

For real $f$, a candidate $u\in U\subset C(K,\mathbb{R})$, and active set $E=\{x:|f(x)-u(x)|=\Vert f-u{\Vert }_{\infty }\}$, the candidate is best precisely when every $v\in U$ has some $x\in E$ with $(f(x)-u(x))v(x)\le 0$. A direction with strictly matching error signs on the whole active set gives a uniformly improving small perturbation by compactness. Conversely one oppositely signed active value already prevents improvement. This criterion yields the Chebyshev alternation theorem.