= Cosine substitution for polynomial approximation
{title2=$\widetilde f(\theta)=f(\cos\theta)$}
Cosine substitution is an isometry from continuous functions on $[-1,1]$ to even continuous $2\pi$-periodic functions. It satisfies $\omega(\widetilde f,\delta)\le\omega(f,\delta)$ 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.
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