A note on some expansions of p-adic functions

Grzegorz Szkibiel. "A note on some expansions of p-adic functions." Acta Arithmetica 61.2 (1992): 129-142. <http://eudml.org/doc/206456>.

@article{GrzegorzSzkibiel1992,
abstract = {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.},
author = {Grzegorz Szkibiel},
journal = {Acta Arithmetica},
keywords = {orthogonal system of p-adic functions; expansions},
language = {eng},
number = {2},
pages = {129-142},
title = {A note on some expansions of p-adic functions},
url = {http://eudml.org/doc/206456},
volume = {61},
year = {1992},
}

TY - JOUR
AU - Grzegorz Szkibiel
TI - A note on some expansions of p-adic functions
JO - Acta Arithmetica
PY - 1992
VL - 61
IS - 2
SP - 129
EP - 142
AB - 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.
LA - eng
KW - orthogonal system of p-adic functions; expansions
UR - http://eudml.org/doc/206456
ER -