Generalized continuation

Abstract

In this paper the operation of analytic continuation is generalized by relaxing the condition that a direct continuation of a function must have the same values as the original on the intersection of their domains of definition. Thus the generalized continuations of a function can have some other property in common with the original function such as being preimages of a single function under a local integral operator. This generalization is accomplished by developing A- continuation of F = ((fα Sα) | fα ϵ ɸ and Sa a ball in Ln} with respect to a collection of maps, A, of subsets of F into F. A must satisfy some compatibility conditions. Many of the proofs in this development parallel those for analytic continuation and lead to the introduction of a manifold on which the generalized continuation is single valued. A generalized continuation of function elements (fα, Sα) is achieved when all the fαs are complex valued functions defined on Sα and some examples are given. © 1972 Pacific Journal of Mathematics.