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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a [[continuous function|0X3WGX95FE]] that is [[periodic|0ATF6Q0633]] with period $2\\pi$, and let $\\varepsilon \u003e 0$.\nThen there is a [[trigonometric polynomial|J3NJQ5FZJ5]] $T$ such that\n$$\\sup_{x \\in \\R} \\abs{f(x) - T(x)} \u003c \\varepsilon.$$\nEquivalently, the trigonometric polynomials are dense in the space $C(\\mathbb{T})$ of continuous functions on $\\mathbb{T} = \\R / 2\\pi\\Z$ with the supremum norm.\n\n## Remarks\n\nOne proof uses the [[Stone–Weierstrass theorem|1P8D3MK79H]] on the compact space $\\mathbb{T}$.\nThe 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 $\\mathbb{T}$.\nAnother proof is [[Fejér's theorem|TPJZCSXSVN]]: the Cesàro means of the Fourier series of $f$ are trigonometric polynomials that converge to $f$ [[uniformly|08VZ57PT26]].\nThe partial sums of the Fourier series of $f$ itself need not converge uniformly, and need not converge at all.\nSo the approximating polynomial is in general not a Fourier partial sum.\nThe 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.\nA consequence is that a continuous $2\\pi$-periodic function whose Fourier coefficients all vanish is identically zero.\n\n## Explanations\n\n### L1 human\n\nEvery continuous periodic function can be imitated, as closely as one likes and at every point at once, by a finite sum of waves.\nThe number of waves needed grows as the required precision grows.\nThis is the periodic version of the Weierstrass theorem that continuous functions on an interval are uniform limits of polynomials.\n\n### L2 human\n\nThe result is the basic density fact of Fourier analysis on the circle.\nSince $C(\\mathbb{T})$ is dense in $L^p(\\mathbb{T})$ for $1 \\le p \u003c \\infty$, the trigonometric polynomials are also dense in these spaces.\nFor $p = 2$ this gives the completeness of the Fourier monomials, so that they form an orthonormal basis of $L^2(\\mathbb{T})$.\nIt also gives uniqueness: two continuous periodic functions with the same Fourier coefficients are equal.\nThe rate of approximation in terms of the modulus of continuity of $f$ is the subject of Jackson's theorems.\n",
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