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1992 | 61 | 2 | 129-142
Tytuł artykułu

A note on some expansions of p-adic functions

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EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
61
Numer
2
Strony
129-142
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-05-21
poprawiono
1991-07-04
Twórcy
  • Institute of Mathematics, Szczecin University, Wielkopolska 15, 70-451 Szczecin, Poland
Bibliografia
  • [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).
  • [2] N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-functions, Springer, New York 1977, 9-36, 91-117.
  • [3] J. Rutkowski, On some expansions of p-adic functions, Acta Arith. 51 (1988), 233-345.
  • [4] W. H. Schikhof, Ultrametric Calculus, Cambridge University Press, 1984.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-aav61i2p129bwm
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