Korovkin-Type Approximation Theorem

• Korovkin-type approximation theorem is a generalization of classical positive operator approximation, ensuring convergence on dense test function sets.
• It extends convergence concepts by incorporating rough weighted ideal and weighted equi-ideal methods, adapting to different convergence modes in normed and locally solid spaces.
• Key properties include closedness, convexity, and strict convexity of limit sets under specific conditions, providing robust convergence guarantees in function approximations.

The Korovkin-type approximation theorem provides a powerful generalization of classical approximation results, characterizing the convergence properties of sequences of positive linear operators on spaces of functions through the convergence behavior on a small set of test functions. Recent developments have extended the framework of Korovkin-type theorems to rough weighted ideal convergence and weighted equi-ideal convergence, significantly broadening the scope of approximation theory in normed spaces and locally solid Riesz spaces (Aziz et al., 21 Dec 2025, Ghosal et al., 2021).

1. Classical Korovkin Approximation and Context

The classical Korovkin theorem states that, for a sequence of positive linear operators $(L_n)$ on $C[a,b]$, uniform convergence of $L_n(f)$ to $f$ for all $f$ is implied by convergence on a finite set of test functions (typically $1$, $x$, $x^2$). This principle underpins much of the theory of positive approximation processes.

Research has introduced generalized modes of convergence—statistical, ideal, and rough convergence—motivating the reframing of Korovkin-type results in abstract settings such as normed spaces, Banach spaces, and Riesz spaces. These directions emphasize convergence determined “almost everywhere” outside small exceptional sets dictated by summability ideals or weights, or up to a prescribed roughness parameter.

2. Rough Weighted Ideal Convergence: Definitions and Key Notions

Let $(X, \|\cdot\|)$ denote a normed space, $\I$ an admissible ideal on $\N$, and $\{\omega_t\}_{t \in \N}$ a sequence of positive weights bounded below by $\beta > 0$. The rough weighted ideal convergence of a sequence $(x_t)$ to $x_*$ of roughness $r \ge 0$ is defined by

$x_t \xrightarrow[r]{(\omega_t,\I)} x_* \iff \forall \epsilon > 0,\ \{t: \omega_t\|x_t - x_*\| > r + \epsilon\} \in \I.$

The rough weighted ideal limit set is

$\mathrm{RWI\!L}^r(x_t) = \{x_* \in X : x_t \xrightarrow[r]{(\omega_t, \I)} x_*\}.$

This notion generalizes classical, statistical, and rough convergence by choice of the ideal and weights, as detailed in (Aziz et al., 21 Dec 2025).

Analogously, for locally solid Riesz spaces $(E, \tau)$ with a neighborhood base $\mathcal{N}$ at $0$ and $V \in \mathcal{N}$ representing the degree of roughness, rough weighted $\mathcal{I}_\tau$-convergence is defined using neighborhoods, with the limit-set $W\mathcal{I}_\tau\text{-}\mathrm{LIM}_V x$ accordingly (Ghosal et al., 2021).

3. Weighted Equi-Ideal Convergence and Korovkin-Type Theorem

The extension to sequences of functions $f_t : F \to X$ (where $X$ is a normed space and $F$ an index set) leads to the concept of weighted equi-ideal convergence. For an analytic $P$-ideal $\I$ on $\N$, $\{f_t\}$ is weighted equi-ideal convergent of degree $r \ge 0$ to $f$ on $F$ if, for all $\epsilon > 0$,

$\bigcap_{y \in F} \{t : \omega_t\|f_t(y) - f(y)\| > r + \epsilon\} \in \I.$

This generalizes equi-statistical convergence (Aziz et al., 21 Dec 2025).

The Korovkin-type approximation theorem in this context asserts: For an appropriately chosen system of test functions $\{f_k\}_{k=1}^m$ generating a subspace dense in $C(F)$ (or analogous structure in $X$), if

$f_t \xrightarrow[r]{\text{weighted equi-ideal}} f$

for each test function $f_k$, then the same convergence holds for every $f$ in a closed subspace generated by the $f_k$. The result serves as a unifying generalization of previous Korovkin-type theorems (e.g., [Theorem 2.4, Karakuş et al.]) and addresses corrections to earlier results (as in [Theorem 2.2, Akdağ, Results Math., 2017]) (Aziz et al., 21 Dec 2025).

4. Structure and Properties of Rough Weighted Limit Sets

Several fundamental structural results for the rough weighted ideal limit set $\mathrm{RWI\!L}^r(x_t)$ include:

• Closedness, Convexity, and Boundedness: The rough limit set is always closed and convex. If $\liminf \omega_t = \alpha > 0$, then $$\mathrm{diam}(\mathrm{RWI\!L}^r(x_t)) \le 2r/\alpha$$. In uniformly convex Banach spaces and for $P$-ideals, the set is strictly convex.
• Topological Complexity: For analytic $P$-ideals, $\mathrm{RWI\!L}^r(x_t)$ is an $F_{\sigma\delta}$ subset of $X$ and hence Borel.
• Minimal Degree and Nonemptiness: The minimal convergent degree $\tilde r(x_t)$ is the infimum of $r$ such that the limit set is nonempty. In reflexive spaces, $\mathrm{RWI\!L}^{\tilde r}(x_t)$ is nonempty; in uniformly convex spaces with a $P$-ideal, it is a singleton.
• Representation of Closed Sets: In separable $X$, every nonempty closed $F \subset X$ can be expressed as the intersection over $r > 0$ of the rough cluster sets for a suitable sequence and ideal.

Illustrative examples demonstrate the necessity of conditions: compactness may fail, strict convexity requires uniform convexity, and the rough limit set can be infinite, closed, yet non-compact (Aziz et al., 21 Dec 2025).

5. Rough Weighted Cluster Points and Maximal Ideals

Associated with each sequence is the set of rough weighted ideal cluster points $(\omega_t, \I)\text{-} \Gamma^r_{x_t}$, defined by the requirement that the set of indices with weighted deviation less than $r+\epsilon$ is not in $\I$ for every $\epsilon > 0$. The limit set always embeds in the cluster set, and for maximal ideals these sets coincide. This yields a maximal-ideal characterization: $\I$ is maximal if and only if the two sets agree for all sequences and $r \ge 0$ (Aziz et al., 21 Dec 2025).

6. Generalizations, Applications, and Counterexamples

The frameworks of rough weighted $\mathcal{I}_\tau$-convergence and weighted equi-ideal convergence subsume earlier results on statistical and rough convergence, and unify several concepts:

• For metric (or normed) spaces with the constant weight and the density-zero ideal, rough convergence and rough statistical convergence are recovered.
• The limit set and cluster set behavior illustrate that closedness, boundedness, and convexity are sensitive to properties of the weights, ideals, and the underlying space. Non-trivial counterexamples demonstrate failure of compactness or convexity, particularly when weights are not $\I$-bounded, or when the underlying space fails appropriate convexity properties (Ghosal et al., 2021).
• The Korovkin-type results have direct implications for the theory of positive operator approximation, offering convergence assurance under substantially weakened regularity assumptions for the sequence and the limiting process (Aziz et al., 21 Dec 2025).

7. Summary Table: Key Properties (Rough Weighted Ideal Limit Sets)

Property	Conditions	Result
Closedness	Always in normed spaces	Limit set is closed
Convexity	Always in normed spaces	Limit set is convex
Strict Convexity	Uniformly convex Banach space, $P$-ideal	Limit set is strictly convex
Singleton	Uniformly convex Banach space, $P$-ideal, $r = \tilde r$	Limit set is a singleton
Topological Complexity	Analytic $P$-ideal	$F_{\sigma\delta}$ subset (Borel)
Maximal-Ideal Coincidence	$\I$ maximal	Limit set equals cluster set

These results establish a flexible and robust foundation for the extension of Korovkin-type approximation theorems beyond classical settings, engaging with rough and ideal-mediated convergence processes to characterize fine-grained operator approximation behavior (Aziz et al., 21 Dec 2025, Ghosal et al., 2021).