# A note on some expansions of p-adic functions

Acta Arithmetica (1992)

- Volume: 61, Issue: 2, page 129-142
- ISSN: 0065-1036

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topGrzegorz 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 -

## References

top- [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). Zbl0692.42009
- [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. Zbl0605.12007
- [4] W. H. Schikhof, Ultrametric Calculus, Cambridge University Press, 1984.

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