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Korovkin type theorems for operators acting on functions of polynomial and exponential growth on $[0,\infty)$

This paper establishes two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions with polynomial or exponential growth on $[0,\infty)$ , demonstrating that pointwise convergence on specific test functions implies convergence for the entire class, with direct applications to classical Baskakov and Szász–Mirakjan operators.

Original authors: Ulrich Abel, Ana-Maria Acu, Margareta Heilmann, Ioan Rasa

Published 2026-06-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ulrich Abel, Ana-Maria Acu, Margareta Heilmann, Ioan Rasa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a machine that takes a picture of a function (a mathematical curve) and tries to recreate it. In the world of mathematics, these machines are called operators. Some of these machines are "positive linear operators," which is a fancy way of saying they are reliable tools that don't distort the basic shape of the picture; they just smooth it out or approximate it.

The big question mathematicians ask is: If this machine gets the picture right for a few simple, basic shapes, will it eventually get the picture right for any complex shape we throw at it?

This paper answers "Yes," but with a specific rulebook depending on how "wild" the complex shape is.

Here is the breakdown of their discovery using simple analogies:

1. The Two Types of "Wild" Functions

The authors look at functions defined on the number line starting from zero ( $[0, \infty)$ ). They divide these functions into two camps based on how fast they grow:

The Polynomial Growers: These are functions that grow like a standard hill or a parabola (e.g., $x^2$ , $x^3$ ). They get big, but they don't explode instantly. Think of these as gentle slopes.
The Exponential Growers: These are functions that grow incredibly fast, like a virus spreading or money in a high-interest bank account (e.g., $e^x$ ). Think of these as steep, vertical cliffs.

2. The "Test Drive" Rule (The Korovkin Theorem)

The paper proves a "Test Drive" rule for these machines.

The Scenario:
Imagine you have a machine ( $L_n$ ) that is supposed to copy a function ( $f$ ). You don't know if it works for every function yet.
The Test: You feed the machine a few specific, simple "test functions" (like $1$, $x$ , and $x^2$ for gentle slopes, or specific exponential curves for the steep cliffs).

The Discovery:

For Gentle Slopes (Polynomials): If the machine learns to perfectly copy the simple polynomial shapes ( $1, x, x^2$ ), the authors prove that it will automatically learn to copy any function that grows like a polynomial, even if that function is very wiggly or complex.
For Steep Cliffs (Exponentials): If the machine learns to perfectly copy the simple exponential shapes ( $e^{\lambda x}$ ), it will automatically learn to copy any function that grows exponentially, no matter how fast it shoots up.

3. How They Proved It (The "Sandwich" Trick)

The authors didn't just guess; they used a clever mathematical "sandwich" technique to prove it.

Imagine you have a mysterious, complex function $f$ that you want to approximate.

The Top Bun: They found a simple polynomial (or exponential) shape that sits above your complex function everywhere.
The Bottom Bun: They found another simple shape that sits below your complex function everywhere.
The Meat: Your complex function is stuck in the middle.

Because the machine is good at copying the "Buns" (the simple shapes), and the machine is "positive" (it doesn't flip things upside down), the machine's copy of the "Meat" gets squeezed tighter and tighter between the copies of the Buns. As the machine gets better at the Buns, it is forced to get better at the Meat.

4. Real-World Examples Mentioned

The paper applies this rule to two famous, classic machines:

The Baskakov Operators: These are great at handling the "Gentle Slopes" (polynomial growth).
The Szász–Mirakjan Operators: These are great at handling the "Steep Cliffs" (exponential growth).

The authors show that because these machines pass the "Test Drive" on simple shapes, they are guaranteed to work for all the complex shapes in their respective categories.

Summary

In short, this paper provides a universal "quality control" checklist. It tells us that if a mathematical approximation tool works on a few basic building blocks, it is mathematically guaranteed to work on a massive, complex class of functions, provided those functions don't grow faster than the tool is designed to handle. It's a way of saying, "If you can build a perfect model of a brick and a beam, you can build a perfect model of the whole house, as long as the house follows the laws of physics (growth limits) we set."

Technical Summary: Korovkin Type Theorems for Operators on Functions of Polynomial and Exponential Growth

Problem Statement
The paper addresses the approximation properties of sequences of positive linear operators, denoted as $L_n$ , acting on continuous functions defined on the half-line $[0, \infty)$ . While classical Korovkin-type theorems establish convergence for functions in specific subspaces (typically polynomials or bounded continuous functions), this work focuses on extending these convergence guarantees to broader classes of unbounded functions. Specifically, the authors investigate conditions under which $L_n f$ converges pointwise to $f$ for functions exhibiting either polynomial growth or exponential growth. The challenge lies in constructing sufficient conditions based on the convergence of $L_n$ on a set of "test functions" that ensure convergence for the entire target class of unbounded functions.

Methodology
The authors employ a constructive approach rooted in the classical Korovkin framework but adapted for unbounded domains and growth conditions. The core methodology involves:

Test Function Selection:
- For polynomial growth, the test set consists of the space $\Pi$ of all polynomial functions.
- For exponential growth, the test set includes exponential functions of the form $\exp_\lambda(x) = e^{\lambda x}$ for all $\lambda \in \mathbb{R}$ .
Constructive Approximation via Lemmas:
The proofs rely on two pivotal lemmas (Lemma 2.2 and Lemma 3.2) that allow the bounding of an arbitrary target function $f$ (satisfying specific growth constraints) by functions within the test space (polynomials or linear combinations of exponentials).
- Polynomial Case: The authors construct a polynomial $P \in \Pi$ that dominates a continuous function $f$ (where $|f| \le p$ for some $p \in \Pi$ ) globally on $[0, \infty)$ while matching $f$ at a specific point $a$ with a controlled error $\epsilon$ . This is achieved by utilizing high-degree polynomials of the form $K(\frac{x-a}{\ell})^{2m}$ to dominate the growth of $f$ at infinity while maintaining local control near $a$ .
- Exponential Case: Similarly, for functions bounded by $A e^{\omega x}$ , the authors construct a function $G$ composed of finite linear combinations of exponentials (derived from $\alpha(x) = \cosh(x) - 1$ ) that dominates $f$ globally and approximates it at a point $a$ within $\epsilon$ .
Squeeze Theorem Application:
Using the positivity and linearity of the operators $L_n$ , the authors sandwich the operator applied to $f$ ( $L_n f$ ) between the operators applied to the constructed upper and lower bounds. Since the operators converge to the identity on the test functions (by assumption), the limits of the upper and lower bounds converge to $f(a) \pm \epsilon$ . By letting $\epsilon \to 0$ , pointwise convergence is established.

Key Contributions and Results
The paper establishes two primary theorems:

Theorem 2.1 (Polynomial Growth): Let $L_n$ be a sequence of positive linear operators on a subspace $D \subset C[0, \infty)$ containing polynomials $\Pi$ . If $L_n p \to p$ pointwise for all $p \in \Pi$ , then $L_n f \to f$ pointwise for any $f \in D$ satisfying $|f(x)| \le p(x)$ for some $p \in \Pi$ .
Theorem 3.1 (Exponential Growth): Let $L_n$ be positive linear operators on a subspace $D$ containing all exponentials $\exp_\lambda$ . If $L_n \exp_\lambda \to \exp_\lambda$ pointwise for all $\lambda \in \mathbb{R}$ , then $L_n f \to f$ pointwise for any $f \in D$ satisfying $|f(x)| \le A \exp(\omega x)$ for constants $A, \omega > 0$ .

Direct Applications
The authors apply these general theorems to two classical operator sequences:

Baskakov Operators ( $V_n$ ): Proven to converge pointwise for functions with polynomial growth.
Szász–Mirakjan Operators ( $S_n$ ): Proven to converge pointwise for functions with exponential growth.
The paper notes that for these operators, the convergence on the respective test functions (polynomials for Baskakov, exponentials for Szász–Mirakjan) is a known result, allowing the new theorems to immediately yield convergence for the broader function classes.

Significance and Claims
The paper claims to offer an elementary framework for establishing pointwise convergence for a broad class of positive linear operators. The significance lies in the method's generality: it does not require complex functional analytic machinery but rather relies on the construction of specific dominating functions within the test space. The authors emphasize that this approach can be applied to various operators beyond the specific examples provided, provided the operators satisfy the necessary convergence on the chosen test functions. The work is modest in scope, focusing strictly on establishing sufficient conditions for pointwise convergence and demonstrating their utility on classical operators, without proposing new experimental applications or future theoretical extensions beyond the immediate mathematical context.