Dini–Lipschitz criterion

Remarks

Let $E_N(f)$ be the uniform distance from $f$ to the trigonometric polynomials of degree at most $N$. Since $S_N T = T$ for such a polynomial $T$, one gets $\sup \abs{f - S_N f} \le (1 + L_N) E_N(f)$, where $L_N$ is the Lebesgue constant. Jackson's theorem gives $E_N(f) \le C \omega(1/N)$ with an absolute constant $C$, and $L_N = O(\log N)$, so the error is $O(\omega(1/N) \log N)$, which tends to $0$ by hypothesis. Example: every Hölder continuous $f$, with $\omega(\delta) \le C \delta^\alpha$ for some $\alpha > 0$, satisfies the hypothesis; so does $f$ with $\omega(\delta) \le C / (\log(1/\delta))^{2}$ for small $\delta$.

Going deeper

For a student

The proof is a model of a general principle: if a linear approximation process has norms $L_N$, then it converges for all functions whose best approximation error is $o(1/L_N)$. Here the norms grow like $\log N$, and Jackson's theorem converts best approximation error into modulus of continuity. The criterion contains the uniform convergence of Fourier series of Hölder and Lipschitz continuous functions.