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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.

Every continuous $2\pi$-periodic function admits degree-at-most-$n$ trigonometric polynomials with supremum norm error bounded by a universal constant times its modulus of continuity at scale $1/n$, for $n\ge 1$. This first-order Jackson-type estimate transfers to algebraic polynomials through cosine substitution for polynomial approximation.

For a periodic function use the modulus of continuityOn the interval, take the supremum over pairs of points whose distance is at most $\delta$. The mean value theorem applied to cosine, whose derivative has absolute value at most one, givesBoth cosine values lie in $[-1,1]$, so for $|h|\le \delta$,Taking the supremum provesThe cosine substitution for polynomial approximation is also a linear isometry in the supremum norm, since cosine maps a full period onto $[-1,1]$; its image consists of even continuous periodic functions.