Uniform Distribution of Sequences and its interplay with Functional Analysis

TL;DR

This paper explores the connections between uniform distribution of sequences and functional analysis, proving new results about Cesàro summability and weak* convergence in Banach spaces and their duals, with implications for convex sets and classical distribution theorems.

Contribution

It introduces novel links between uniform distribution concepts and functional analysis, extending classical theorems to functions of bounded variation.

Findings

Sequences in Banach spaces can be Cesàro summable to points in the closed convex hull.

Sequences in dual spaces can have arithmetic means that weak* converge to certain functionals.

Generalization of a classical uniform distribution theorem to functions of bounded variation.

Abstract

In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in Uniform Distribution itself. So let $E$ be a Banach space. Then we prove:\\ (a) If $F$ is a bounded subset of $E$ and $x \in \overline{\co}(F)$ (= the closed convex hull of $F$), then there is a sequence $(x_n) \subseteq F$ which is Ces\`{a}ro summable to $x$.\\ (b) If $E$ is separable, $F \subseteq E^*$ bounded and $f \in \overline{\co}^{w^*}(F)$, then there is a sequence $(f_n) \subseteq F$ whose sequence of arithmetic means $\frac{f_1+\dots+f_N}{N}$, $N \ge 1$ weak$^*$-converges to $f$. By the aid of the Krein-Milman theorem, both (a) and (b) have interesting implications for closed, convex and bounded subsets $\Omega$ of $E$ such that $\Omega=\overline{\co}(\ex \Omega)$ and for weak$^*$ compact and convex subsets of $E^*$. Of particular interest is the case when $\Omega=B_{C(K)^*}$, where $K$ is a compact metric space. By further expanding the previous ideas and results, we are able to generalize a classical theorem of Uniform Distribution which is valid for increasing functions $\varphi:I=[0,1] \rightarrow \mathbb{R}$ with $\varphi(0)=0$ and $\varphi(1)=1$, for functions $\varphi$ of bounded variation on $I$ with $\varphi(0)=0$ and total variation $V_0^1 \varphi=1$.