In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by A(T), is the space of absolutely convergent Fourier series.
On the other hand an integration by parts, together with the Cauchy–Schwarz inequality and Parseval's formula, shows that More generally, for
Wiener (1932, 1933) proved that if f has absolutely convergent Fourier series and is never zero, then its reciprocal 1/f also has an absolutely convergent Fourier series.
Many other proofs have appeared since then, including an elementary one by Newman (1975).
Gelfand (1941, 1941b) used the theory of Banach algebras that he developed to show that the maximal ideals of A(T) are of the form which is equivalent to Wiener's theorem.