On the Sum of Two Periodic Functions

Abstract

Carl G. Townsend is a State Department exchange professor, currently lecturing at Rangoon Arts and Sciences University. Last year he was a lecturer on a FullbrightHays Grant at Rajshoki University in East Pakistan. He maintains his position as Associate Professor at Southern Illinois University at Carbondale as well. It is well known (Olmsted, 1959, p. 550) that the set of all real-valued periodic functions with common domain R (the real number system) do not constitute a vector space. For example, the two functions f and g defined by f(x) = sin x and g(x) = sin cx, where c is a real constant, are periodic, but their sum f + g, defined by (f + g) (x) = f(x) + g(x), is periodic if and only if c is rational. It is also well known (Olmsted, 1959, p. 549) that if f is any real-valued function on R, then the set P of periods of f is an additive group and therefore contains 0 only (in case f is not periodic), is dense in R, or consists of all integral multiples of a positive number p known as the least positive period of f Furthermore, any nonconstant periodic function either is everywhere discontinuous or has a least positive period. A simple example of a nonconstant function with a dense set of periods is the characteristic function of the set Q of all rational numbers (with values 1 on Q and 0 on the complement of Q), whose set of periods is Q itself. The general problem to which we are addressing ourselves in this note is represented by such questions as the following: Under what conditions is the sum of two periodic functions periodic? What role does boundedness play? What role does continuity play? What about the special case where the two functions have incommensurable least positive periods (that is, least positive periods with an irrational ratio)? We present some answers to a few such questions, and believe that some of these results contain an element of the unexpected. Many unanswered questions remain, and a few of these are formulated below in specific terms and presented in