Jackson-type estimate

A Jackson-type estimate bounds an approximation error by a smoothness quantity such as a modulus of continuity, with the smoothness evaluated at a scale comparable to the reciprocal of the approximating degree.

For $p\ge 1$ and $f\in C^p({\mathbb{T}}^n)$, there is a real trigonometric polynomial with coordinate frequencies $|k_j|\le N$ whose uniform error is at most $C_{n,p}{N}^{-p}\Vert f{\Vert }_{C^p}$. One obtains the multivariable statement by applying bounded one-dimensional Jackson approximation operators successively in each coordinate and telescoping their errors. Each operator commutes with differentiation in the other coordinates, and its uniform operator norm is bounded independently of $N$; the total error is bounded by a constant times $N^{-p}{\sum }_j\Vert {\partial }_j^{p}f{\Vert }_{\infty }$.

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.

The normalized even kernelis nonnegative, has integral one, and is a trigonometric polynomial of degree $2(n-1)$. Its mass is concentrated on a scale $1/n$: $J_n(t)\le C\min (n,n^{-3}|t{|}^{-4})$ for $|t|\le \pi$. This controls a weighted second moment and produces the Jackson operator estimate.

For $j_{n}f(x)={\int }_{-\pi }^{\pi }f(x-t)J_n(t)dt$, evenness and unit mass reduce the error to an integral of second differences. The Jackson kernel bound and the scaling inequality for the second modulus of smoothness yield $\Vert j_{n}f-f{\Vert }_{\infty }\le C{\omega }_2(f,1/n)$. Since the degree is $2(n-1)$, a degree-$N$ estimate uses $n=\lfloor N/2\rfloor +1$. For $f\in C^2$ this gives $E_N(f)\le C^{\prime }{N}^{-2}\Vert f^{\prime \prime }{\Vert }_{\infty }$.