Fejér's theorem

Remarks

The proof writes $\sigma_N f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(t) f(x - t) \, dt$ with the Fejér kernel $F_N$. Since $F_N \ge 0$ and $F_N$ has mean $1$, the error $\sigma_N f(x) - f(x)$ is a weighted average of $f(x - t) - f(x)$. For small $\abs{t}$ this is small by uniform continuity of $f$, and for $\abs{t} \ge \delta$ the weight $F_N(t)$ is uniformly small for large $N$.

Example: the theorem shows that trigonometric polynomials are uniformly dense in the continuous $2\pi$-periodic functions, since each $\sigma_N f$ is one. It also shows that two continuous periodic functions with the same Fourier coefficients are equal.

Continuity is needed for uniform convergence: each $\sigma_N f$ is continuous, so a uniform limit of them is continuous, and a function with a jump cannot be such a limit. The theorem cannot be improved to the partial sums $S_n f$ themselves: there are continuous periodic functions whose Fourier series diverges at a point.

Going deeper

For a student

Fejér's theorem is the model case of summability methods applied to Fourier series. Its proof uses only three properties of the Fejér kernel: positivity, unit mean and concentration at $0$. Any kernel with these properties gives a similar theorem, such as Abel–Poisson summation with the Poisson kernel. Versions for integrable functions say that $\sigma_N f \to f$ in mean and at every point of continuity.