P-Adic Mikusinski calculus and its applications to the fourier and the mahler expansions of locally constant functions

Abstract

We consider the p-adic counterpart of Mikusinski’s operational calculus based on the algebra C(ℤp) of continuous functions on ℤp taking values in ℂp and equipped with the discrete Laplace convolution. Elements of the field (hyperfunctions) corresponding to shift operators, difference operators, and the indefinite sum operator are considered. A notion of p-adic exponent is generalized. Applications to the Fourier and the Mahler expansions of the indicator function of a ball and the convolution of two indicator functions are provided. Two ways of applying the p-adic analog of Mikusinski’s operational calculus lead us to the Fourier expansion for the fractional part of a p-adic number.