Jackson's inequality

Remarks

Proof idea for $r = 0$: approximate $f$ by its convolution with the Jackson kernel $J_m$ of degree at most $n$. The kernel is nonnegative with mean $1$, so $\abs{f - f * J_m} \le \frac{1}{2\pi} \int \omega(f, \abs{t}) J_m(t) \, dt$. With $\omega(f, \abs{t}) \le (1 + n\abs{t}) \, \omega(f, 1/n)$ and $\frac{1}{2\pi} \int \abs{t} J_m(t) \, dt = O(1/n)$, the bound follows. The case $r \ge 1$ follows by induction on $r$, using $E_n(f) \le \frac{C}{n} E_n(f')$ for continuously differentiable $2\pi$-periodic $f$.

Example: if $f$ is Lipschitz with constant $L$, then $\omega(f, 1/n) \le L/n$, so $E_n(f) \le C_0 L / n$.

The same bounds hold for algebraic polynomials of degree at most $n$ on $[-1, 1]$ when $r = 0$, through the substitution $x = \cos \theta$. For $\abs{f^{(r)}} \le 1$ with $r \ge 1$, the sharp bound is $E_n(f) \le K_r / (n + 1)^r$, where $K_r$ is the Favard constant, by Favard's theorem.

Going deeper

For a student

This is the basic direct theorem of approximation theory. Its converse, Bernstein's inverse theorem, shows that the rates it gives are the right ones when $0 < \alpha < 1$ in the Hölder scale. Together with Bernstein's inequality, it underlies the characterisation of smoothness classes by approximation rates. For interpolation and Fourier partial sums, it combines with Lebesgue constants to give convergence results such as the Dini–Lipschitz test.