Real analytic calculus for several variables and applications (Q2047435)

Let \(A\) be a commutative, unital complex Banach algebra with involution. If \(U\) is an open subset of \(\mathbb R^{2n}\), the algebra \(A(U)\) of real analytic functions in \(2n\) real variables is algebraically isomorphic to an inductive limit of holomorphic function algebras. Using this and holomorphic functional calculus, a real analytic functional calculus is obtained for \(n\)-tuples of elements of \(A\). This functional calculus yields analogues of the classical theorems of Wiener and Lévy on absolutely convergent Fourier series in certain weighted algebras.