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- Complex positive definite functions on stripsPublication . Buescu, Jorge; Coelho, Jose; Symeonides, A.We characterize a holomorphic positive definite function f defined on a horizontal strip of the complex plane as the Fourier–Laplace transform of a unique exponentially finite measure on R. With this characterization, the classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become respectively the real and imaginary sections of the corresponding complex integral representation. We provide minimal holomorphy assumptions for this characterization and derive conclusions for meromorphic functions under minimal positive definiteness conditions. Further characterizations are derived from conditions on the derivatives of f arising from the study of the usual concepts of moment, moment-generating function and characteristic function in this context.
- Positive-definiteness and integral representations for special functionsPublication . Buescu, Jorge; Paixão, AntónioIt is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the Gamma, zeta and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.
- The measure transition problem for meromorphic polar functionsPublication . Buescu, Jorge; Paixão, AntónioIn 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.