Weighted Equi-Ideal Convergence

• Weighted equi-ideal convergence is a generalized mode that unifies statistical and weighted convergence through analytic P-ideals and structured weight sequences.
• The methodology uses weights exceeding a positive threshold and lower-semicontinuous submeasures to control large deviations in function sequences.
• It underpins advanced Korovkin-type approximation theorems, ensuring uniform convergence while addressing anomalies from operator-induced irregularities.

Weighted equi-ideal convergence is a generalized mode of convergence for sequences of functions, unifying and extending various notions of statistical and weighted convergence through the framework of analytic $P$-ideals and sequences of weights. It provides a refined analytic approach, particularly relevant in functional analysis and approximation theory, and plays a central role in advanced versions of Korovkin-type approximation theorems (Aziz et al., 21 Dec 2025).

1. Formal Definition and Foundational Components

Let $I_{(\varphi)}$ denote an analytic $P$-ideal on $\mathbb{N}$ generated by a lower-semicontinuous submeasure $\varphi$. Consider a sequence of weights $\{\omega_t\}_{t\in\mathbb{N}}$ with $\omega_t>\beta>0$ for all $t$, and let $f_t, f\in C(K)$, where $K\subset\mathbb{R}$ is compact. The sequence $\{f_t\}$ is said to converge to $f$ in the sense of weighted equi-ideal convergence (abbreviated as $\omega$–equi–$I_{(\varphi)}$–convergence) if, for every $\varepsilon>0$, the functions

$$h_{j,\varepsilon}(x)=\varphi\left(\left\{t\in\mathbb{N}:\omega_t|f_t(x)-f(x)|>\varepsilon\right\}\setminus[1,j]\right), \quad x\in K,$$

satisfy

$$\lim_{j\to\infty}\sup_{x\in K}h_{j,\varepsilon}(x)=0.$$

This condition quantifies the vanishing, outside finite sets, of the $\varphi$-mass of indices with large weighted deviations from $f$, uniformly over $K$.

2. Structure of Weights and Ideals

The convergence notion crucially depends on the properties of both the weights and the ideal:

• Weights: $\omega_t>\beta>0$ for all $t$ is required. Frequently, the weight sequence is assumed $I_{(\varphi)}$–bounded, i.e., $\exists\mu>0$ such that $\{t:\omega_t>\mu\}\in I_{(\varphi)}$.
• Ideals: $I_{(\varphi)}$ is analytic and necessarily of the form $\operatorname{Exh}(\varphi)$, where $\varphi$ is a lower-semicontinuous submeasure. The analytic $P$-ideal property ensures regularity and closure under countable unions, which is fundamental for the uniform convergence criteria employed.

3. Relationship to Statistical and Weighted Statistical Convergence

Weighted equi-ideal convergence generalizes previous convergence modes:

• Equi-statistical convergence: For $\omega_t\equiv 1$ and $\varphi(A)=\sup_n |A\cap[1,n]|/n$, $I_{(\varphi)}=I_\delta$ (density-zero ideal). In this case, $\omega$–equi–$I_{(\varphi)}$–convergence coincides with ordinary equi-statistical convergence as in Balcerzak–Dems–Komisarski.
• Weighted equi-statistical convergence: For $\omega_t>0$ and $\varphi(A)=\sup_n |A\cap [1,\theta_n]|/\theta_n$, $\theta_n=\sum_{t\leq n}\omega_t$, this recovers the framework of Akdağ.

This unification clarifies both the scope and the limitations of prior formulations: weighted equi-ideal convergence encompasses both density-based ($I_\delta$) and weighted density-based ideals, but also enables convergence analysis with respect to any analytic $P$-ideal.

4. Borel Structure and Monotonicity Properties

For analytic $P$-ideals, the set of rough $I_{(\varphi)}$-limits of a sequence in a normed space is always an $F_{\sigma\delta}$ set, and hence Borel [(Aziz et al., 21 Dec 2025), Prop. 2.1]. This regularity property underpins the descriptive set-theoretic rigor of the convergence concept. Furthermore, the mode is monotonic in roughness: if $L_t(f;x)$ converges to $f$ in the $\omega$–equi–$I_{(\varphi)}$ sense with some roughness parameter $r_1$, it does so with any larger $r_2>r_1$.

5. Korovkin-Type Approximation: Generalized Theorem and Proof Structure

The formulation of a Korovkin-type theorem using weighted equi-ideal convergence achieves both a generalization and a correction of prior results (Aziz et al., 21 Dec 2025):

Theorem:

Let $I_{(\varphi)}$ be an analytic $P$-ideal and $\{\omega_t\}$ $I_{(\varphi)}$–bounded. For compact $K\subset\mathbb{R}$ and a sequence of positive linear operators $L_t:C(K)\to C(K)$, the following are equivalent:

• (a) $L_t(f)\to f$ in the sense of $\omega$–equi–$I_{(\varphi)}$–convergence for all $f\in C(K)$,
• (b) $L_t(e_i)\to e_i$ ($\omega$–equi–$I_{(\varphi)}$) on $K$ for $i=0,1,2$, with $e_0(x)=1$, $e_1(x)=x$, $e_2(x)=x^2$.

Proof Sketch:

The implication (a)$\Rightarrow$(b) is immediate, while (b)$\Rightarrow$(a) proceeds via:

1. Local uniform continuity of $f$ to reduce estimation of $L_t(f;x)-f(x)$ to that of the canonical test functions.
2. Control of large deviations via weighted inequalities involving the three test functions.
3. Application of the $\omega$–equi–$I_{(\varphi)}$–convergence on the test functions to conclude convergence for all of $C(K)$.

This result corrects deficiencies in prior attempts by accounting for anomalies induced by pathological modifications (e.g., artificial "spikes" in operator images).

6. Illustrative Examples and Limitations

Three representative examples demonstrate the generality and necessity of the $\omega$–equi–$I_{(\varphi)}$ formulation:

Example	Specialization	Outcome
6.1	$\omega_t\equiv 1$, $I_\delta$	Ordinary equi-statistical convergence
6.2	Weighted $\omega_t$, weighted $I_\delta$	Weighted equi-statistical convergence
6.3	Bernstein operator + "spike"	Classical criterion fails (no convergence)

Specifically, Example 6.3 shows that requiring mere weighted equi-statistical convergence for test functions is insufficient when operators introduce extraneous local oscillations; the full analytic ideal-bounded formulation is necessary for the Korovkin theorem to hold robustly. A plausible implication is that analytic $P$-ideal boundedness imposes adequate control over the weight-induced exceptions to preserve uniform Korovkin-type approximation.

7. Significance and Context within Approximation Theory

Weighted equi-ideal convergence establishes a unified mathematical infrastructure for analyzing convergence properties of operator sequences under broad weighting and ideal constraints. It generalizes and corrects earlier results in Korovkin-type approximation, ensuring that the classical three-function test criterion is retained even in this highly generalized setting (Aziz et al., 21 Dec 2025). The approach is intrinsically related to descriptive set theory via the $F_{\sigma\delta}$ property, and connects the analytic study of rough and weighted cluster points with practical approximation theorems. This framework is significant for the rigorous analysis of functional approximation under nonstandard averaging, density, and weighting regimes.