Generalizations of Dini's Theorem under Weakened Monotonicity Conditions

TL;DR

This paper extends Dini's Theorem by replacing strict monotonicity with more flexible conditions such as equicontinuity and convexity, broadening its applicability to convergence of function sequences.

Contribution

It introduces new theorems that generalize Dini's result under weakened monotonicity assumptions, including equicontinuity and controlled variation.

Findings

Generalized convergence criteria for function sequences

Broader conditions ensuring uniform convergence

Enhanced understanding of convergence under relaxed assumptions

Abstract

Dini's Theorem guarantees that a monotone sequence of continuous functions converges pointwise on a compact interval to a continuous limit that converges uniformly. In this paper, we establish new theorems generalizing Dini's result by replacing the restrictive monotonicity assumption with more flexible conditions like equicontinuity, convexity, and controlled variation hypotheses.