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  "body": "\n## Definition\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$, and for an integrable function $f$ on $\\mathbb{T}$ let $\\hat f(n)$, $n \\in \\Z$, be its [[Fourier coefficients|5NN2YZZR3V]].\nThe *Wiener algebra* is the set\n$$A(\\mathbb{T}) = \\Big\\{ f \\colon \\mathbb{T} \\to \\C \\text{ continuous} : \\sum_{n \\in \\Z} \\abs{\\hat f(n)} \u003c \\infty \\Big\\},$$\nof [[continuous|0X3WGX95FE]] functions with an [[absolutely convergent Fourier series|HHBGB5R5S2]], with pointwise addition and multiplication and the norm\n$$\\norm{f}_A = \\sum_{n \\in \\Z} \\abs{\\hat f(n)}.$$\n\n## Remarks\n\nThe map $f \\mapsto (\\hat f(n))_{n \\in \\Z}$ is an isometric isomorphism of $A(\\mathbb{T})$ onto $\\ell^1(\\Z)$.\nIt turns pointwise products into convolutions: $\\widehat{fg}(n) = \\sum_{k \\in \\Z} \\hat f(k) \\hat g(n - k)$.\nSo $\\norm{fg}_A \\le \\norm{f}_A \\norm{g}_A$, and $A(\\mathbb{T})$ is a commutative unital [[Banach algebra|9KYWN94QDN]].\nEvery $f \\in A(\\mathbb{T})$ satisfies $\\sup_x \\abs{f(x)} \\le \\norm{f}_A$.\n\nExample: $f(x) = 2 + \\cos x$ lies in $A(\\mathbb{T})$ with $\\hat f(0) = 2$, $\\hat f(\\pm 1) = 1/2$, and $\\norm{f}_A = 3$.\n\nNon-example: $f(x) = \\sum_{n \\ge 2} \\frac{\\sin nx}{n \\log n}$.\nIts coefficients decrease to $0$ and $n \\cdot \\frac{1}{n \\log n} \\to 0$, so by the Chaundy–Jolliffe theorem the series converges uniformly and $f$ is continuous.\nBut $\\abs{\\hat f(\\pm n)} = \\frac{1}{2 n \\log n}$ and $\\sum_{n \\ge 2} \\frac{1}{n \\log n} = \\infty$, so $f \\notin A(\\mathbb{T})$.\n\nThe characters of $A(\\mathbb{T})$ are the evaluations $f \\mapsto f(x)$ at points $x \\in \\mathbb{T}$; this is the key to [[Wiener's lemma|NC768TWZ3D]].\n",
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