Generalized Analytic Continuation

Abstract

One of the most important notions in connection with the study of analytic functions is that of analytic continuation. Several recent investigations have encountered situations where there is associated with an analytic function in a domain, another function analytic in a contiguous domain which can lay claim to being a “continuation” of the original function, even though the original function is nowhere continuable in the classical sense. Such a situation arises, for instance, if the first function has nontangential limiting values almost everywhere on some smooth arc of the boundary and the second function has identical nontangential limiting values almost everywhere on that arc. As follows from a theorem of Lusin and Privalov, the two functions then uniquely determine one another.