The extension problem in free harmonic analysis

This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus the concept a positive definite function group and our primary goal is to understand how such functions can be extended from balls finite radius entire group. Previous work showed that extensions always exist we study problem simultaneous extension multiple functions. More specifically, define ‘relative energy’ which measures proximity between pair functions, show ball without increasing their relative energy. The proof analytic, involving differentiation noncommutative Szegö parameters.