Statistically p-Upward Quasi-Cauchy Sequences and Cone-Valued Continuity

TL;DR

This paper introduces and studies statistically p-upward quasi-Cauchy sequences, characterizing their compactness and continuity properties, and explores their implications in function spaces and approximation theory.

Contribution

It defines statistically p-upward quasi-Cauchy sequences, characterizes their compactness and continuity, and analyzes their role in function spaces and approximation applications.

Findings

Statistically p-upward compact sets are characterized by lower boundedness.

Statistically p-upward continuity implies uniform continuity on bounded below sets.

The function space SUC_p(E) is a closed convex cone but not a subspace.

Abstract

We introduce statistically $p$-upward quasi-Cauchy sequences, defined by the condition $\lim_{n\to\infty}\frac{1}{n}|\{k\leq n: x_k - x_{k+p}\geq\varepsilon\}|=0$ for every $\varepsilon>0$, and develop the corresponding notions of compactness and continuity. We prove that a subset of $\mathbb{R}$ is statistically $p$-upward compact if and only if it is bounded below, characterizing lower boundedness sequentially. Statistically $p$-upward continuity is shown to imply uniform continuity on below bounded sets. The function space $\mathrm{SUC}_p(E)$ is a closed convex cone that fails to be a vector subspace -- distinguishing it from all previously studied sequential continuity spaces. We establish that every non-decreasing uniformly continuous function belongs to $\mathrm{SUC}_p(E)$, use Weyl's equidistribution theorem to show $\sin x\notin\mathrm{SUC}_p(\mathbb{R})$, prove a step-parameter hierarchy, and show that $\mathrm{SUC}_p(E)\cap C_b(E)$ is nowhere dense in $C_b(E)$. As an application, we develop a one-sided error control theory for function approximation, illustrated by Bernstein operators on a pharmacokinetic model. The inclusion relations among the continuity types studied and open problems are provided.