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  "body": "\n## Statement\n\nLet $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].\nLet $f \\colon \\mathbb{T} \\to \\C$ be a function with an absolutely convergent Fourier series, that is, $f(z) = \\sum_{n \\in \\Z} c_n z^n$ with $\\sum_{n \\in \\Z} \\abs{c_n} \u003c \\infty$.\nIf $f(z) \\ne 0$ for every $z \\in \\mathbb{T}$, then $1/f$ also has an absolutely convergent Fourier series.\n\n## Remarks\n\nThe result is also called Wiener's $1/f$ theorem.\nProof idea in Gelfand theory: the functions with absolutely convergent Fourier series form the Wiener algebra, isomorphic to the Banach algebra $\\ell^1(\\Z)$ with convolution.\nIts [[characters|M7JVFJ3MMX]] are the evaluations at points of $\\mathbb{T}$, and its [[Gelfand transform|VWNSHKJT92]] is $f$ itself.\nAn element of a commutative unital Banach algebra is invertible if and only if no character vanishes on it.\nSince $f$ has no zero on $\\mathbb{T}$, $f$ is invertible in the Wiener algebra, and its inverse is $1/f$.\n\nExample: $f(z) = 2 + z$ has no zero on $\\mathbb{T}$, and indeed $1/f(z) = \\tfrac{1}{2} \\sum_{n \\ge 0} (-z/2)^n$ has absolutely summable coefficients.\n\nThe hypothesis is needed: $f(z) = 1 + z$ has absolutely convergent Fourier series but vanishes at $z = -1$, and $1/f$ is unbounded near $-1$, so it has no absolutely convergent Fourier series.\n\n## Explanations\n\n### L1 human\n\nSome periodic functions are so nice that the sum of the sizes of their Fourier coefficients is finite.\nWiener's lemma says that if such a function never takes the value $0$, then its reciprocal is just as nice.\nThis is surprising at first: nothing obvious controls the Fourier coefficients of $1/f$.\n\n### L2 human\n\nWiener's original proof was by direct estimates; Gelfand's proof with characters made the theory of commutative Banach algebras famous.\nThe same argument proves analogues for other Banach algebras of functions, and the lemma is a key step in Wiener's Tauberian theorem.\nA function with an absolutely convergent Fourier series is continuous.\nThe lemma says that the invertible elements of the Wiener algebra are exactly its functions without zeros on the circle.\n",
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