Let {Xi. i ≥1} be a sequence of independent exponential random variables with mean 1. Setting S(n) = X1 +…+ Xn, S(0) = 0, we define

where f is a real-valued, measurable function satisfying

.

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

KMS states of a twisted convolution algebra of Schwartz functions on a vector group are classified and related to KMS states of twisted L1-algebras for certain subquotients. The KMS states for the subquotient algebras are also related to Fock states of vector groups. In the particular case of the subquotient Tn × ℤn of ℚ2n this links the Fock space construction of the theta functions with their appearance in KMS states of loop groups and in the Kac character formula.

A non-singular solution of the bosonic SU(1, 1)/U(1) no-scale supergravity field equations is constructed for an anisotropic cosmological model which is of Bianchi type VIIh.