First Jackson theorem for periodic approximation
= First Jackson theorem for periodic approximation
{title2=$E_n^{\mathrm{trig}}(f)\le C\omega(f,1/n)$}
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 $n\ge1$. This first-order <Jackson-type estimate> transfers to algebraic polynomials through <cosine substitution for polynomial approximation>.