The Wiener Lemma and Certain of Its Generalizations

Let T be the unit circle in the complex plane. Let $/ be the Banach algebra of all complex valued continuous functions on T with absolutely convergent Fourier series. Norbert Wiener [10] proved that if any ƒ in sf is invertible in the ring of continuous functions on T, then \/f also is an element of sf . Paul Levy [7] generalized Wiener's result, showing that for each ƒ in sf and each complex analytic function O that is defined on a neighborhood of f(T), $(ƒ) belongs to stf . Levy did so by an argument that shows, more generally, that 0(ƒ) belongs to sf whenever ƒ is in J / and O is analytic on some neighborhood of f(T) in C . Later, G. E. Silov [9] established such a result for a class of Banach algebras of continuous functions that includes sf . Silov's proof uses the Cauchy-Weil integral formula for an analytic function of several complex variables.