Bochner’s theorem on Fourier-Stieltjes integrals in the framework of quaternion analysis

Abstract

Let σ(x) be a nondecreasing function, such that σ(−∞) = 0,σ(∞) = 1 and let us denote by B the class of functions which can be represented by a Fourier-Stieltjes integral f(t) = ∫ −∞∞eitxdσ(x). In continuation to [12], we prove a generalization of the classical theorem of Bochner on Fourier integral transforms to quaternion functions belonging to a subclass of B. The underlying functions are continuous functions of bounded variation defined in R2 and taking values on the quaternion algebra. Additionally, we introduce the definition of convolution of quaternion functions of bounded variation.