The measure transition problem for meromorphic polar functions

In very general conditions, meromorphic polar functions (i.e. functions exhibiting some kind of positive or co-positive definiteness) separate the complex plane into horizontal or vertical strips of holomophy and polarity, in each of which they are characterized as integral transforms of exponentially finite measures. These measures characterize both the function and the strip. We study the problem of transition between different holomorphy strips, proving a transition formula which relates the measures on neighbouring strips of polarity. The general transition problem is further complicated by the fact that a function may lose polarity upon strip crossing and in general we cannot expect polarity, or even some specific related form of integral representation, to exist. We show that, even in these cases, a relevant analytical role will be played by exponentially finite signed measures, which we construct and study. Applications to especially significant examples like the Gamma, zeta or Bessel functions are performed.