{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T23:39:18Z","timestamp":1772581158253,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T00:00:00Z","timestamp":1743033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007414","name":"Deanship of Graduate Studies and Scientific Research at Qassim University","doi-asserted-by":"publisher","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}],"id":[{"id":"10.13039\/501100007414","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This research focuses on the approximation properties of Kantorovich-type operators using Frobenius\u2013Euler\u2013Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence.<\/jats:p>","DOI":"10.3390\/axioms14040252","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T04:11:40Z","timestamp":1743135100000},"page":"252","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Approximation Results: Sz\u00e1sz\u2013Kantorovich Operators Enhanced by Frobenius\u2013Euler\u2013Type Polynomials"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5681-9563","authenticated-orcid":false,"given":"Nadeem","family":"Rao","sequence":"first","affiliation":[{"name":"Department of Mathematics, University Center for Research and Development, Chandigarh University, Mohali 140413, Punjab, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-3509","authenticated-orcid":false,"given":"Mohammad","family":"Farid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7584-4214","authenticated-orcid":false,"given":"Mohd","family":"Raiz","sequence":"additional","affiliation":[{"name":"Department of Applied Science and Humanities, Global Institute of Technology and Management, 5 KM Mile Stone, Haily Mandi Road, Kheda Khurampur, Farrukhnagar, Gurugramn 122506, Haryana, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,27]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"D\u00e9monstration du th\u00e9or\u00e8me de Weierstrass fond\u00e9e sur le calcul de probabilit\u00e9s","volume":"13","author":"Bernstein","year":"1913","journal-title":"Commun. 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