A note on some expansions of p-adic functions

Tom 61 / 1992

Grzegorz Szkibiel Acta Arithmetica 61 (1992), 129-142 DOI: 10.4064/aa-61-2-129-142


Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by $(ϕₘ)_{m∈ ℕ₀}$. The system $(ϕₘ)_{m∈ ℕ₀}$ 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 $(ϕₘ)_{m∈ ℕ₀}$. This paper is a remark to Rutkowski's paper. We define another system $(hₙ)_{n∈ ℕ₀}$ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system $(hₙ)_{n∈ ℕ₀}$ 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 $(hₙ)_{n∈ ℕ₀}$ than to $(ϕₘ)_{m∈ ℕ₀}$. Moreover, a function in C(ℤₚ,ℂₚ) has an expansion with respect to $(hₙ)_{n∈ ℕ₀}$ if it has an expansion with respect to $(ϕₘ)_{m∈ ℕ₀}$. At the end of this paper an example is given of a function which has an expansion with respect to $(hₙ)_{n∈ ℕ₀}$ but not with respect to $(ϕₘ)_{m∈ ℕ₀}$. 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.


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