A note on some expansions of p-adic functions

Grzegorz Szkibiel

Acta Arithmetica, Tome 62 (1992), p. 129-142 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1992-01-01

Zbl 0726.11075

EUDML-ID : urn:eudml:doc:206456

Bibliographie

[000] [1] B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh Series and Walsh Transforms. Theory and Applications, Nauka, Moscow 1987, 9-41 (in Russian). | Zbl 0692.42009

[001] [2] N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-functions, Springer, New York 1977, 9-36, 91-117.

[002] [3] J. Rutkowski, On some expansions of p-adic functions, Acta Arith. 51 (1988), 233-345. | Zbl 0605.12007

[003] [4] W. H. Schikhof, Ultrametric Calculus, Cambridge University Press, 1984.