Several functional equations defined on groups arising from stochastic distance measures.

Abstract

SEVERAL FUNCTIONAL EQUATIONS DEFINED ON GROUPS ARISING FROM STOCHASTIC DISTANCE MEASURES Heather B. Hunt June 25th, 2014 Several functional equations related to stochastic distance measures have been widely studied when defined on the real line. This dissertation generalizes several of those results to functions defined on groups and fields. Specifically, we consider when the domain is an arbitrary group, G, and the range is the field of complex numbers, C. We begin by looking at the linear functional equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) for all p, q, r, s ∈ G. The general solution f : G×G→ C is given along with a few specific examples. Several generalizations of this equation are also considered and used to determine the general solution f, g, h, k : G×G→ C of the functional equation f(pr, qs) + g(ps, qr) = h(p, q) + k(r, s) for all p, q, r, s ∈ G. We then consider the non-linear functional equation f(pr, qs) + f(ps, qr) = f(p, q) f(r, s). The solution f : G×G→ C is given for all p, q, r, s ∈ G when f is an abelian function. It is followed by the structure of the general solution, f , dependent upon how the function acts on the center of the group. Several generalizations of the equation are also considered. The general structure of the solution f, g, h : G×G→ C of the functional equation f(pr, qs) + f(ps, qr) = g(p, q)h(r, s) is given for all p, q, r, s ∈ G, dependent upon how the function h acts on the center of the group. Future plans related to these equations will be given.