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  "body": "\n## Summary\n\nThe Wiener algebra $A(\\mathbb{T})$ is the set of continuous periodic functions with an [[absolutely convergent Fourier series|HHBGB5R5S2]].\nIts norm is the sum of the absolute values of the Fourier coefficients.\nWith pointwise multiplication it is a commutative [[Banach algebra|9KYWN94QDN]], isometrically isomorphic to the sequence algebra $\\ell^1(\\Z)$ with convolution.\nIt is the setting of [[Wiener's lemma|NC768TWZ3D]] and a basic example in the Gelfand theory of commutative Banach algebras.\n\n## Explanations\n\n### L1 human\n\nTake all periodic functions whose Fourier coefficients have a finite sum of absolute values.\nAdding or multiplying two such functions gives another one, so they form an algebra.\nMeasuring a function by the total size of its coefficients makes this algebra complete.\nThe functions $\\cos x$ and $\\sum_{n \\in \\Z} 2^{-\\abs{n}} e^{inx}$ belong to it, while the square wave does not, because it is not even continuous.\n",
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