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A note on some expansions of p-adic functions

100%

Szkibiel G.

Acta Arithmetica

1992

tom 61

nr 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.

Controlling a non-homogeneous Timoshenko beam with the aid of the torque

63%

Sklyar G., Szkibiel G.

International Journal of Applied Mathematics and Computer Science

2013

tom 23

nr 3

587-598

Considered is the control and stabilizability of a slowly rotating non-homogeneous Timoshenko beam with the aid of a torque. It turns out that the beam is (approximately) controllable with the aid of the torque if and only if it is (approximately) controllable. However, the controllability problem appears to be a side-effect while studying the stabilizability. To build a stabilizing control one needs to go through the methods of correcting the operators with functionals so that they have finally the appropriate form and the results on C⁰-continuous semigroups may be applied.

Ograniczanie wyników

1 Acta Arithmetica

1 International Journal of Applied Mathematics and Computer Science

2 Szkibiel G.

1 Sklyar G.

1 2013

1 1992

Wyniki wyszukiwania

A note on some expansions of p-adic functions

Controlling a non-homogeneous Timoshenko beam with the aid of the torque