Density of trigonometric polynomials in continuous functions

Remarks

One proof uses the Stone–Weierstrass theorem on the compact space $\mathbb{T}$. The trigonometric polynomials form a subalgebra of $C(\mathbb{T})$ that contains the constants, is closed under complex conjugation, and separates points, since $e^{ix}$ is injective on $\mathbb{T}$. Another proof is Fejér's theorem: the Cesàro means of the Fourier series of $f$ are trigonometric polynomials that converge to $f$ uniformly. The partial sums of the Fourier series of $f$ itself need not converge uniformly, and need not converge at all. So the approximating polynomial is in general not a Fourier partial sum. The periodicity hypothesis is necessary: every trigonometric polynomial $T$ satisfies $T(-\pi) = T(\pi)$, so the function $x$ on $[-\pi, \pi]$ cannot be approximated uniformly on that interval. A consequence is that a continuous $2\pi$-periodic function whose Fourier coefficients all vanish is identically zero.

Going deeper

For a student

The result is the basic density fact of Fourier analysis on the circle. Since $C(\mathbb{T})$ is dense in $L^p(\mathbb{T})$ for $1 \le p < \infty$, the trigonometric polynomials are also dense in these spaces. For $p = 2$ this gives the completeness of the Fourier monomials, so that they form an orthonormal basis of $L^2(\mathbb{T})$. It also gives uniqueness: two continuous periodic functions with the same Fourier coefficients are equal. The rate of approximation in terms of the modulus of continuity of $f$ is the subject of Jackson's theorems.