A note on some expansions of p-adic functions

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by ${\displaystyle {(\varphi ₘ)}_{m\in \mathbb{N}₀}}$. The system ${\displaystyle {(\varphi ₘ)}_{m\in \mathbb{N}₀}}$ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski has also considered some questions concerning expansions of functions from C(ℤₚ,ℂₚ) with respect to ${\displaystyle {(\varphi ₘ)}_{m\in \mathbb{N}₀}}$. This paper is a remark to Rutkowski's paper. We define another system ${\displaystyle {(hₙ)}_{n\in \mathbb{N}₀}}$ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system ${\displaystyle {(hₙ)}_{n\in \mathbb{N}₀}}$ can be viewed as a p-adic analogue of the well-known Haar system of real functions (see [1]). It turns out that in general functions are expanded much easier with respect to ${\displaystyle {(hₙ)}_{n\in \mathbb{N}₀}}$ than to ${\displaystyle {(\varphi ₘ)}_{m\in \mathbb{N}₀}}$. Moreover, a function in C(ℤₚ,ℂₚ) has an expansion with respect to ${\displaystyle {(hₙ)}_{n\in \mathbb{N}₀}}$ if it has an expansion with respect to ${\displaystyle {(\varphi ₘ)}_{m\in \mathbb{N}₀}}$. At the end of this paper an example is given of a function which has an expansion with respect to ${\displaystyle {(hₙ)}_{n\in \mathbb{N}₀}}$ but not with respect to ${\displaystyle {(\varphi ₘ)}_{m\in \mathbb{N}₀}}$. Throughout the paper the ring of p-adic integers, the field of p-adic numbers and the completion of its algebraic closure will be denoted by ℤₚ, ℚₚ and ℂₚ respectively (p prime). In addition, we write ℕ₀= ℕ ∪ {0} and E={0,1,...,p-1}. The author would like to thank Jerzy Rutkowski for fruitful comments and remarks that permitted an improvement of the presentation.