### A method for constructing orthonormal bases for non-archimedean Banach spaces of continuous functions

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Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$. The system ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$ 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 ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$. This paper is a remark to Rutkowski’s paper. We define another system ${\left(h\u2099\right)}_{n\in \mathbb{N}\u2080}$ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system...

We study dynamical systems in the non-Archimedean number fields (i.e. fields with non-Archimedean valuation). The main results are obtained for the fields of p-adic numbers and complex p-adic numbers. Already the simplest p-adic dynamical systems have a very rich structure. There exist attractors, Siegel disks and cycles. There also appear new structures such as fuzzy cycles. A prime number p plays the role of parameter of a dynamical system. The behavior of the iterations depends on this parameter...

A detailed study of power series on the Levi-Civita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and re-expandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and...

This paper is mainly concerned with extensions of the so-called Vishik functional calculus for analytic bounded linear operators to a class of unbounded linear operators on ${c}_{0}$. For that, our first task consists of introducing a new class of linear operators denoted $W\left({c}_{0}(J,\omega ,\mathbb{K})\right)$ and next we make extensive use of such a new class along with the concept of convergence in the sense of resolvents to construct a functional calculus for a large class of unbounded linear operators.

In this paper, following the $p$-adic integration theory worked out by A. F. Monna and T. A. Springer [4, 5] and generalized by A. C. M. van Rooij and W. H. Schikhof [6, 7] for the spaces which are not $\sigma $-compacts, we study the class of integrable $p$-adic functions with respect to Bernoulli measures of rank $1$. Among these measures, we characterize those which are invertible and we give their inverse in the form of series.

We study geodesic completeness for left-invariant Lorentz metrics on solvable Lie groups.