An isomorphism of Paley-Wiener type (Q579564)

An isometric isomorphism is established between the Smirnov space \(F_ 2(\Omega)\) in a convex N-gon \(\Omega\subset {\mathbb{C}}\) and the space which consists of N-tuples \((u_ 0,...,u_{N-1})\) of entire functions \(u_ k\) of exponential type (the type of \(u_ k\) is defined by the geometry of \(\Omega)\) belonging to \(L_ 2({\mathbb{R}})\) and satisfying some linear coupling condition. This isomorphism maps every function \(u\in E_ 2(\Omega)\) to the N-tuple of Fourier transforms of its boundary values on the sides of \(\Omega\).