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

Ana-Maria Acu; Ioan Rasa; Margareta Heilmann; Ulrich Abel

arxiv: 2606.27517 · v1 · pith:LNKG5WFXnew · submitted 2026-06-25 · 🧮 math.NA · cs.NA

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

Ulrich Abel , Ana-Maria Acu , Margareta Heilmann , Ioan Rasa This is my paper

Pith reviewed 2026-06-29 01:04 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Korovkin theoremspositive linear operatorsapproximation on [0,∞)Baskakov operatorsSzász-Mirakjan operatorspolynomial growthexponential growthpointwise convergence

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The pith

Positive linear operators converging pointwise on test functions also converge on all continuous functions of polynomial or exponential growth on [0,∞).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves two Korovkin-type theorems for sequences of positive linear operators on continuous functions defined on the half-line. If the operators converge pointwise on a suitable collection of test functions, then the same holds for every function whose growth is at most polynomial or exponential. The proofs are elementary and yield convergence results for the classical Baskakov and Szász-Mirakjan operators as immediate consequences. The framework applies to any positive linear operators satisfying the initial test-function hypothesis.

Core claim

Under the assumption of pointwise convergence on suitable test functions, sequences of positive linear operators converge pointwise on the entire class of continuous functions with polynomial or exponential growth on [0,∞).

What carries the argument

Korovkin-type test-function sets that control polynomial and exponential growth classes, allowing extension of pointwise convergence from the test set to all functions in the growth classes.

If this is right

Pointwise convergence holds for the Baskakov operators on all continuous functions of polynomial growth.
Pointwise convergence holds for the Szász-Mirakjan operators on all continuous functions of exponential growth.
The same test-function method yields convergence for any other positive linear operators that satisfy the initial hypothesis.
The results cover the full scale of growth rates between polynomial and exponential without additional assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to other unbounded intervals or to operators defined on spaces with different growth weights.
Numerical schemes based on these operators could now be analyzed for approximation error on solutions that grow at most exponentially.
Similar test-function arguments might apply to convergence in weighted norms rather than pointwise.

Load-bearing premise

The operators are positive and linear, and the pointwise convergence assumption holds on a set of test functions sufficient to dominate the polynomial and exponential growth.

What would settle it

A concrete positive linear operator that converges pointwise on the chosen test functions yet fails to converge at some point for a continuous function of polynomial growth on [0,∞).

read the original abstract

We prove two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on $[0,\infty)$. Under the assumption of pointwise convergence on suitable test functions, we establish pointwise convergence for all functions with polynomial or exponential growth. As direct applications, we obtain convergence results for the classical Baskakov and Sz\'asz--Mirakjan operators. The proposed method offers an elementary framework that can be applied to a broad class of positive linear operators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Korovkin theorems to polynomial and exponential growth on [0,∞) via test-function assumptions, with direct applications to Baskakov and Szász-Mirakjan operators.

read the letter

The punchline is that they prove Korovkin-type theorems for positive linear operators on [0,∞) that work for functions with polynomial or exponential growth, given convergence on test functions, and then get results for Baskakov and Szász-Mirakjan operators as applications.

What is new here is the extension to these specific growth classes on the half-line. Most Korovkin work stays with bounded functions or compact domains, so this broadens the setting a bit. The elementary framework they propose could be reusable for other operators.

The paper does well by making the applications direct, so you see the test functions actually work for those classical cases without extra effort. The central claim is conditional on the test set being rich enough, which is standard but needs to hold.

The soft spot is minor: the sufficiency of the test functions for controlling the growth is the key step, and one would check the details in the proofs to see if the estimates are clean. No sign of circular reasoning or invented entities from the abstract.

This paper is for approximation theorists working with positive operators. Someone looking for extensions of Korovkin theorems to unbounded intervals would get value from the concrete examples.

It deserves a serious referee because the result is a clean, targeted extension with verifiable applications.

I would recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves two Korovkin-type theorems for sequences of positive linear operators on continuous functions on [0,∞). Under the assumption of pointwise convergence on a suitable set of test functions, the results establish pointwise convergence for all functions with polynomial or exponential growth. Direct applications are derived for the Baskakov and Szász-Mirakjan operators, and the approach is presented as an elementary framework applicable to a broad class of such operators.

Significance. If the derivations hold, the work supplies a conditional but elementary extension of classical Korovkin theory to growth-controlled function classes on unbounded intervals. The explicit applications to two standard operators and the framing as a reusable method constitute concrete strengths for approximation theory.

minor comments (3)

[Abstract] The abstract and introduction should explicitly name the test-function sets used for the polynomial-growth and exponential-growth cases (e.g., {1, x, x²} or {e^{-x}, xe^{-x}, …}) so that the sufficiency claim can be checked at a glance.
[§2] Notation for the growth classes (polynomial of degree ≤ n versus exponential of order ≤ α) should be introduced once in §2 and used consistently thereafter; current usage mixes “functions of polynomial growth” with “functions f such that |f(x)| ≤ C(1+x)^k”.
[Theorem 3.1 and Theorem 3.2] The statements of the two main theorems should include the precise pointwise-convergence hypothesis on the test set as a numbered assumption rather than an inline sentence, to make the logical structure transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no individual points to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a conditional Korovkin-type result: pointwise convergence on a chosen test set (sufficient to control polynomial/exponential growth) implies the same for the full target class of functions, under positivity and linearity. This is a direct extension via standard approximation arguments and does not reduce any claimed prediction or theorem to its own inputs by definition, fitting, or self-citation chain. No load-bearing self-citations, ansatzes, or renamings are indicated in the provided structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of positive linear operators and the classical Korovkin framework; no free parameters or invented entities are mentioned.

axioms (2)

standard math Positive linear operators preserve positivity and linearity on continuous functions
Invoked as background for the Korovkin-type setting
domain assumption Pointwise convergence on a suitable finite set of test functions implies convergence on the target growth classes
This is the load-bearing hypothesis stated in the abstract

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

M. E. H. Ismail, C. P. May, On a family of approximation operators, J. Math. Anal. Appl., 63 (1978), pp. 446–462

1978
[2]

The complete asymptotic expansion for Baskakov operators

Zhang, C., Wang, Q. The complete asymptotic expansion for Baskakov operators. Analysis in Theory and Applications 23, 76–82 (2007)

2007

[1] [1]

M. E. H. Ismail, C. P. May, On a family of approximation operators, J. Math. Anal. Appl., 63 (1978), pp. 446–462

1978

[2] [2]

The complete asymptotic expansion for Baskakov operators

Zhang, C., Wang, Q. The complete asymptotic expansion for Baskakov operators. Analysis in Theory and Applications 23, 76–82 (2007)

2007