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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.
- Propagation of regularity and positive definiteness: a constructive approachPublication . Buescu, Jorge; Paixão, António; Oliveira, ClaudemirWe show that, for positive de finite kernels, ifspecific forms of regularity (continuity, S-n-differentiability or holomorphy) hold locally on the diagonal, then they must hold globally on the whole domain of positive-definiteness. This local-toglobal propagation of regularity is constructively shown to be a consequence of the algebraic structure induced by the non-negativity of the associated bilinear forms up to order 5. Consequences of these results for topological groups and for positive definite and exponentially convex functions are explored.