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Open AccessArticle

Weighted Approximation by Szász–Mirakyan–Durrmeyer Operators Reproducing Exponential Functions

by Gülsüm Ulusoy Ada and Ali Aral

Mathematics 2026, 14(1), 59; https://doi.org/10.3390/math14010059 - 24 Dec 2025

We examine a Szász–Mirakyan–Durrmeyer type operator that reproduces the functions 1 and

e^{2 a x}

for a fixed parameter

a > 0 .

While its exponential reproduction property has been described in the classical literature, the effect of exponential weights on its [...] Read more.

We examine a Szász–Mirakyan–Durrmeyer type operator that reproduces the functions 1 and

e^{2 a x}

for a fixed parameter

a > 0 .

While its exponential reproduction property has been described in the classical literature, the effect of exponential weights on its approximation behavior has not been studied. In this work, we provide a detailed analysis of the operator in weighted spaces and show that combining exponential reproduction with weighted norms improves the approximation behavior for exponentially growing functions. We also prove that the corresponding sequence of operator norms remains uniformly bounded for a family of exponential weights, ensuring the stability of the operators in the weighted framework. Moreover, we establish new Korovkin-type approximation theorems involving weighted convergence and obtain sharp uniform error estimates in the presence of exponential weights. These results extend the classical theory to weighted exponential settings and highlight several quantitative features that do not arise in the classical case. Full article

(This article belongs to the Special Issue Advances in Operator Theory and Nonlinear Evolution Equations)

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19 pages, 792 KB

Open AccessArticle

Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives

by Vijay Gupta, Ana Maria Acu and Hari Mohan Srivastava

Symmetry 2020, 12(6), 915; https://doi.org/10.3390/sym12060915 - 2 Jun 2020

Cited by 27 | Viewed by 2922

Abstract

In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupaş–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper. Full article

(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)

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