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20 pages, 466 KB

Open AccessArticle

Weighted Approximation by Szász–Mirakyan-Type Operators Preserving Two Exponential Functions

by Gülsüm Ulusoy Ada and Ali Aral

Mathematics 2026, 14(8), 1371; https://doi.org/10.3390/math14081371 - 19 Apr 2026

Viewed by 433

In this paper, we study the weighted approximation properties of a family of Szász–Mirakyan-type operators preserving two exponential functions on the unbounded interval

[0, \infty)

. The operators act on exponential weighted spaces and are analyzed within the framework of [...] Read more.

In this paper, we study the weighted approximation properties of a family of Szász–Mirakyan-type operators preserving two exponential functions on the unbounded interval

[0, \infty)

. The operators act on exponential weighted spaces and are analyzed within the framework of positive linear operator theory. We first establish their well-definedness and boundedness between suitable weighted spaces. By applying a weighted Korovkin-type theorem, we prove convergence in the corresponding weighted norm. Furthermore, we obtain quantitative estimates in terms of a weighted modulus of continuity and derive an order of convergence result. A Voronovskaya-type asymptotic formula is also established, describing the precise asymptotic behavior of the operators. Numerical examples are included to support the theoretical results. Full article

(This article belongs to the Special Issue Recent Developments in Theoretical and Applied Mathematics)

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15 pages, 289 KB

Open AccessArticle

A New Family of Szász–Mirakyan-Type Operators Preserving Two Exponential Functions

by Gülsüm Ulusoy Ada and Ali Aral

Mathematics 2026, 14(7), 1214; https://doi.org/10.3390/math14071214 - 4 Apr 2026

Cited by 2 | Viewed by 551

Abstract

This paper introduces a new family of Szász–Mirakyan-type operators defined by a convex combination of two Poisson-type constructions. The operators preserve the constant function and provide a continuous transition between different exponential behaviors through a parameter sequence. Basic properties of the operators are [...] Read more.

This paper introduces a new family of Szász–Mirakyan-type operators defined by a convex combination of two Poisson-type constructions. The operators preserve the constant function and provide a continuous transition between different exponential behaviors through a parameter sequence. Basic properties of the operators are studied, including the preservation of exponential test functions and the behavior of the first and second central moments. Voronovskaja-type asymptotic results are obtained, describing the effect of the parameter on the asymptotic structure. Moreover, a necessary condition for faster-than

1 / n

approximation is derived. The behavior of the operators is examined through computational evidence, which also confirms the theoretical findings. Full article

(This article belongs to the Special Issue New Advances in Functional Analysis and PDEs)

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13 pages, 318 KB

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

Cited by 1 | Viewed by 870

Abstract

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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24 pages, 2666 KB

Open AccessArticle

The Adaptive Backstepping Synchronization Control for a Kind of Variable-Order Fractional Uncertain Nonlinear Systems

by Xinyu Si, Kangkang Zhang, Jingfei Jiang, Juan L. G. Guirao and Hongkui Li

Fractal Fract. 2025, 9(12), 761; https://doi.org/10.3390/fractalfract9120761 - 24 Nov 2025

Viewed by 815

Abstract

This paper is concerned with the adaptive backstepping synchronization control for a class of variable-order (VO) fractional uncertain nonlinear system with external disturbances and a dead zone. A kind of VO fractional command filter is employed to cope with “explosion of complexity”. The [...] Read more.

This paper is concerned with the adaptive backstepping synchronization control for a class of variable-order (VO) fractional uncertain nonlinear system with external disturbances and a dead zone. A kind of VO fractional command filter is employed to cope with “explosion of complexity”. The unknown nonlinear term in considered system is decomposed into unknown parameters and error functions by the Szász–Mirakyan operator theory. A VO fractional disturbance observer derived in this paper is used to simultaneously deal with the difficulties brought by dead zone and external disturbances. Thus, a VO fractional backstepping synchronization controller with adaptive laws for the system handled is proposed; moreover, the stability of system controlled is established. Finally, numerical examples are given to validate the theoretical results. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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11 pages, 1027 KB

Open AccessArticle

Some Probabilistic Generalizations of the Cheney–Sharma and Bernstein Approximation Operators

by Seng Huat Ong, Choung Min Ng, Hong Keat Yap and Hari Mohan Srivastava

Axioms 2022, 11(10), 537; https://doi.org/10.3390/axioms11100537 - 8 Oct 2022

Cited by 8 | Viewed by 2340

Abstract

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. [...] Read more.

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. It is also shown that the Cheney–Sharma operator is the Szász–Mirakyan operator averaged by a certain probability distribution. Full article

(This article belongs to the Special Issue Mathematical Tools and Techniques Applicable to Probability Theory and Statistics)

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19 pages, 360 KB

Open AccessArticle

A Generalization of Szász–Mirakyan Operators Based on α Non-Negative Parameter

by Khursheed J. Ansari and Fuat Usta

Symmetry 2022, 14(8), 1596; https://doi.org/10.3390/sym14081596 - 3 Aug 2022

Cited by 21 | Viewed by 2501

Abstract

The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say

α

. This new family of Szász–Mirakyan operators is crucial in that it includes both the existing Szász–Mirakyan operator and allows [...] Read more.

The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say

α

. This new family of Szász–Mirakyan operators is crucial in that it includes both the existing Szász–Mirakyan operator and allows the construction of new operators for different values of

α

. Then, the convergence properties of the new operators with the aid of the Popoviciu–Bohman–Korovkin theorem-type property are presented. The Voronovskaja-type theorem and rate of convergence are provided in a detailed proof. Furthermore, with the help of the classical modulus of continuity, we deduce an upper bound for the error of the new operator. In addition to these, in order to show that the convex or monotonic functions produced convex or monotonic operators, we obtain shape-preserving properties of the new family of Szász–Mirakyan operators. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Moreover, we compare this operator with its classical correspondence to show that the new one has superior properties. Finally, some numerical illustrative examples are presented to strengthen our theoretical results. Full article

(This article belongs to the Special Issue Advances in Matrix Transformations, Operators and Symmetry)

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9 pages, 270 KB

Open AccessArticle

Yet Another New Variant of Szász–Mirakyan Operator

by Ana Maria Acu and Gancho Tachev

Symmetry 2021, 13(11), 2018; https://doi.org/10.3390/sym13112018 - 25 Oct 2021

Cited by 14 | Viewed by 2765

Abstract

In this paper, we construct a new variant of the classical Szász–Mirakyan operators,

M_{n}

, which fixes the functions 1 and

e^{a x}, x \geq 0, a \in R

. For these operators, we provide a quantitative Voronovskaya-type result. [...] Read more.

In this paper, we construct a new variant of the classical Szász–Mirakyan operators,

M_{n}

, which fixes the functions 1 and

e^{a x}, x \geq 0, a \in R

. For these operators, we provide a quantitative Voronovskaya-type result. The uniform weighted convergence of

M_{n}

and a direct quantitative estimate are obtained. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Our results improve and extend similar ones on this topic, established in the last decade by many authors. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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 29 | Viewed by 3281

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 [...] Read more.

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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