Wiener algebra
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by , is the space of absolutely convergent Fourier series. Here denotes the circle group. Banach algebra structure. The norm of a function is given by where is the th Fourier coefficient of . The Wiener algebra is closed under pointwise multiplication of functions. Indeed, therefore Thus the Wiener algebra is a commutative unitary Banach algebra. Also, is isomorphic to the Banach algebra , with the isomorphism given by the Fourier transform. Properties. The sum of an absolutely convergent Fourier series is continuous, so where is the ring of  continuous functions on the unit circle. On the other hand an integration by parts, together with the Cauchy–Schwarz inequality and Parseval's formula, shows that More generally, for formula_8 (see ). Wiener's 1/"f" theorem. proved that if has absolutely convergent Fourier series and is never zero, then its reciprocal also has an absolutely convergent Fourier series. Many other proofs have appeared since then, including an elementary one by . which is equivalent to Wiener's theorem.