Wiener's lemma

Remarks

The result is also called Wiener's $1/f$ theorem. Proof 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. Its characters are the evaluations at points of $\mathbb{T}$, and its Gelfand transform is $f$ itself. An element of a commutative unital Banach algebra is invertible if and only if no character vanishes on it. Since $f$ has no zero on $\mathbb{T}$, $f$ is invertible in the Wiener algebra, and its inverse is $1/f$.

Example: $f(z) = 2 + z$ 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.

The 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.

Going deeper

For a student

Wiener's original proof was by direct estimates; Gelfand's proof with characters made the theory of commutative Banach algebras famous. The same argument proves analogues for other Banach algebras of functions, and the lemma is a key step in Wiener's Tauberian theorem. A function with an absolutely convergent Fourier series is continuous. The lemma says that the invertible elements of the Wiener algebra are exactly its functions without zeros on the circle.