Korovkin type theorems for operators acting on functions of polynomial and exponential growth on [0,∞)
Abstract
We establish two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on [0,∞). The first result shows that pointwise convergence on all polynomial test functions implies pointwise convergence for every function in the domain that is dominated by a polynomial. The second result provides an analogous criterion for functions of exponential growth, where the test functions are the exponentials eλx, λ∈ℝ. The results are applied to several classical and modified positive linear operators, including Baskakov, Szász--Mirakjan, Phillips, Post--Widder, Gamma, and Durrmeyer-type operators. Further applications are given to operators associated with modified Bessel functions and hyperbolic functions. The approach provides a unified framework for deriving pointwise convergence on unbounded intervals from convergence on suitable test functions.
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BibTeX
@misc{abel2026korovkin,
title = {{Korovkin type theorems for operators acting on functions of polynomial and exponential growth on $[0,\infty)$}},
author = {Ulrich Abel and Ana-Maria Acu and Margareta Heilmann and Ioan Rasa},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.27517},
eprint = {2606.27517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27517},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Korovkin type theorems for operators acting on functions of polynomial and exponential growth on [0,∞)
AU - Ulrich Abel
AU - Ana-Maria Acu
AU - Margareta Heilmann
AU - Ioan Rasa
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2606.27517
UR - https://arxiv.org/abs/2606.27517
AB - We establish two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on [0,∞). The first result shows that pointwise convergence on all polynomial test functions implies pointwise convergence for every function in the domain that is dominated by a polynomial. The second result provides an analogous criterion for functions of exponential growth, where the test functions are the exponentials e^λx, λ∈ℝ. The results are applied to several classical and modified positive linear operators, including Baskakov, Szász--Mirakjan, Phillips, Post--Widder, Gamma, and Durrmeyer-type operators. Further applications are given to operators associated with modified Bessel functions and hyperbolic functions. The approach provides a unified framework for deriving pointwise convergence on unbounded intervals from convergence on suitable test functions.
ER -