Rough Weighted Ideal Convergence and Korovkin-Type Approximation via weighted equi-ideal convergence

TL;DR

This paper introduces rough weighted ideal convergence and weighted equi-ideal convergence in normed spaces, providing new characterizations, properties, and a Korovkin-type approximation theorem that generalizes and corrects prior results.

Contribution

It develops the concepts of rough weighted ideal limit sets and weighted equi-ideal convergence, extending Korovkin approximation theory with new generalizations and corrections.

Findings

Characterization of maximal ideals in normed spaces

Representation of closed sets via rough weighted ideal limit sets

Establishment of a Korovkin-type approximation theorem for weighted equi-ideal convergence

Abstract

If $\omega_t > \beta$ for every $t \in \mathbb{N}$ and for some $\beta > 0$, then the sequence $\{\omega_t\}_{t \in \mathbb{N}}$ represents a weighted sequence of real numbers. In this article, we primarily introduce the concepts of rough weighted ideal limit set and rough weighted ideal cluster points set associated with sequences in normed spaces. Building on these concepts, we derive several important results, including a characterization of maximal ideals, a representation of closed sets in normed spaces, and an analysis of the minimal convergent degree required for the rough weighted ideal limit set to be non-empty. Furthermore, we demonstrate that for an analytic $P$-ideal, the rough weighted ideal limit set forms an $F_{\sigma\delta}$ subset of the normed space. Finally, we introduce the concept of weighted equi-ideal convergence for sequences of functions with respect to analytic $P$-ideals, extending the notion of equi-statistical convergence [Balcerzak et al., J. Math. Anal. Appl. {328} (1) (2007)]. As an application of this notion, we establish a Korovkin-type approximation theorem that serves both as a generalization of [Theorem 2.4, Karaku{\c{s}} et al., J. Math. Anal. Appl. {339} (2) (2008)] and a correction to [Theorem 2.2, Akda\u{g}, Results Math. {72} (3) (2017)].