= Endpoint obstruction to an algebraic inverse approximation theorem
{title2=$E_n(\sqrt{1-x^2})=O(n^{-1}),\quad\omega(\sqrt{1-x^2},n^{-1})\gtrsim n^{-1/2}$}
The <first Jackson theorem for periodic approximation> applied to $|\sin\theta|$ gives algebraic best error $O(n^{-1})$ for $\sqrt{1-x^2}$. Its ordinary interval <modulus of continuity> at $1/n$ is at least $\sqrt{2/n-1/n^2}$. Substituting that error rate into the unmodified <inverse theorem for trigonometric approximation> would instead bound the modulus by $O(\log(n+1)/n)$, a contradiction. Algebraic inverse estimates must incorporate the endpoint compression of <cosine substitution for polynomial approximation>.
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