Operational calculus on the ring of p-adic integers

Abstract

We develop a one-dimensional Mikusinski type Operational Calculus on the ring of p-adic integers ℤ p . The Calculus utilizes the space C(ℤ p ) of continuous functions with values in the field of p-adic numbers ℚ p . C(ℤ p ) is equipped with a convolution, which turns it into an integral domain. A hyperfunction is by definition an element of a fraction field of C(ℤ p ). We refer to this fraction field as the p-adic Mikusinski field, prove its separability and incompleteness. Hyperfunctions corresponding to operations of shift, taking difference, indefinite summation, differentiation and integration are considered. A generalization of the p-adic exponential function a x is constructed for all a∈ℚ p . Also a variant of the Mikusinski field is considered that allows to treat continuous functions and p-adic measures simultaneously.