Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials
This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e., <inline-...
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2025-05-01
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| author | Nadeem Rao Mohammad Farid Shivani Bansal |
| author_facet | Nadeem Rao Mohammad Farid Shivani Bansal |
| author_sort | Nadeem Rao |
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| description | This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Furthermore, estimates in view of test functions and central moments are studied. Next, rate of convergence is discussed with the aid of the Korovkin theorem and the Voronovskaja type theorem. Moreover, direct approximation results in terms of modulus of continuity of first- and second-order, Peetre’s K-functional, Lipschitz type space, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>r</mi><mrow><mi>t</mi><mi>h</mi></mrow></msup></semantics></math></inline-formula>-order Lipschitz type maximal functions are investigated. In the subsequent section, we present weighted approximation results, and statistical approximation theorems are discussed. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically. |
| format | Article |
| id | doaj-art-a15b37cb9b6d4e3c94072b18a65ee2fe |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-a15b37cb9b6d4e3c94072b18a65ee2fe2025-08-20T03:32:27ZengMDPI AGAxioms2075-16802025-05-0114641810.3390/axioms14060418Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type PolynomialsNadeem Rao0Mohammad Farid1Shivani Bansal2Department of Mathematics, University Center for Research and Development, Chandigarh University, Mohali 140413, Punjab, IndiaDepartment of Mathematics, College of Science, Qassim University, Saudi ArabiaDepartment of Mathematics, University Institute of Sciences, Chandigarh University, Mohali 140413, Punjab, IndiaThis manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Furthermore, estimates in view of test functions and central moments are studied. Next, rate of convergence is discussed with the aid of the Korovkin theorem and the Voronovskaja type theorem. Moreover, direct approximation results in terms of modulus of continuity of first- and second-order, Peetre’s K-functional, Lipschitz type space, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>r</mi><mrow><mi>t</mi><mi>h</mi></mrow></msup></semantics></math></inline-formula>-order Lipschitz type maximal functions are investigated. In the subsequent section, we present weighted approximation results, and statistical approximation theorems are discussed. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically.https://www.mdpi.com/2075-1680/14/6/418modulus of smoothnessmathematical operatorsSzász operatorFrobenius–Euler–Simsek-type polynomialsapproximation algorithmsorder of approximation |
| spellingShingle | Nadeem Rao Mohammad Farid Shivani Bansal Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials Axioms modulus of smoothness mathematical operators Szász operator Frobenius–Euler–Simsek-type polynomials approximation algorithms order of approximation |
| title | Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials |
| title_full | Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials |
| title_fullStr | Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials |
| title_full_unstemmed | Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials |
| title_short | Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials |
| title_sort | szasz beta operators linking frobenius euler simsek type polynomials |
| topic | modulus of smoothness mathematical operators Szász operator Frobenius–Euler–Simsek-type polynomials approximation algorithms order of approximation |
| url | https://www.mdpi.com/2075-1680/14/6/418 |
| work_keys_str_mv | AT nadeemrao szaszbetaoperatorslinkingfrobeniuseulersimsektypepolynomials AT mohammadfarid szaszbetaoperatorslinkingfrobeniuseulersimsektypepolynomials AT shivanibansal szaszbetaoperatorslinkingfrobeniuseulersimsektypepolynomials |