arXiv subjects

Adrian Fellhauer

Publications and source records attributed to Adrian Fellhauer.

3 recordsLinked to original sources

arXiv2018

On the relation of three theorems of analysis to the axiom of choice

In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzelà--Ascoli theorem and the Fréchet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of real numbers. Secondly, some progress is made towards determining the amount of axioms that have to be added to the Zermelo--Fraenkel system so that the uniform boundedness principle holds.

math.LO↗

arXiv2016

Approximation of smooth functions using Bernstein polynomials in multiple variables

In this survey, we use (more or less) elementary means to establish the well-known result that for any given smooth multivariate function, the respective multivariate Bernstein polynomials converge to that function in all derivatives on each compact set. We then go on to strengthen that result to obtain that any smooth function on $\mathbb R^d$ may be approximated locally uniformly in all derivatives by \emph{one} sequence of polynomials. We will use neither the axiom of choice nor the power set axiom. We will use the method of proof by contradiction.

math.CA↗

arXiv2014

On the relation between continuous functions in two different metric spaces

Let the metric space $\mathbb R^n \setminus \sim$ be the metric space of $n$-sized unordered tuples of real numbers. In the following, it will be shown that if a function $φ: \mathbb R^m \to \mathbb R^n \setminus \sim$ is continuous, then there is a continuous function $f: \mathbb R^m \to \mathbb R^n$ such that a natural embedding of $f$ into $\mathbb R^n \setminus \sim$ is equal to $φ$. This theorem is wrong in the complex case. A counterexample is given in [1].

math.MG↗