A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter
The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/8/1596 |
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| author | Khursheed J. Ansari Fuat Usta |
| author_facet | Khursheed J. Ansari Fuat Usta |
| author_sort | Khursheed J. Ansari |
| collection | DOAJ |
| description | The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. 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. |
| first_indexed | 2024-03-09T03:46:13Z |
| format | Article |
| id | doaj.art-2faf03b5b368428e861082f7c83c7466 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T03:46:13Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2faf03b5b368428e861082f7c83c74662023-12-03T14:33:02ZengMDPI AGSymmetry2073-89942022-08-01148159610.3390/sym14081596A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative ParameterKhursheed J. Ansari0Fuat Usta1Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi ArabiaDepartment of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, TurkeyThe main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. 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.https://www.mdpi.com/2073-8994/14/8/1596Szász–Mirakyan operatorsmodulus of continuityVoronovskaja theoremKorovkin-type theoremshape-preserving approximation |
| spellingShingle | Khursheed J. Ansari Fuat Usta A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter Symmetry Szász–Mirakyan operators modulus of continuity Voronovskaja theorem Korovkin-type theorem shape-preserving approximation |
| title | A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter |
| title_full | A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter |
| title_fullStr | A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter |
| title_full_unstemmed | A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter |
| title_short | A Generalization of Szász–Mirakyan Operators Based on <i>α</i> Non-Negative Parameter |
| title_sort | generalization of szasz mirakyan operators based on i α i non negative parameter |
| topic | Szász–Mirakyan operators modulus of continuity Voronovskaja theorem Korovkin-type theorem shape-preserving approximation |
| url | https://www.mdpi.com/2073-8994/14/8/1596 |
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