Limits of integrals involving almost periodic functions (Q1773444)

The author is concerned with the existence of some limits involving almost periodic functions, establishing their existence and asymptotic formulas. For instance it is shown that \[ \int^\infty_1 f(x)\sin(Rx){dx\over x}=- {\ln\| R\|\over 2[R]^2}+ o(1), \] with \([R]\) representing the largest integer in \(R\). Results similar to those related to the integral which defines the mean value for AP functions are also investigated.