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First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.<\/jats:p>","DOI":"10.3390\/axioms12010005","type":"journal-article","created":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T02:31:11Z","timestamp":1671676271000},"page":"5","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Approximation Properties of the Blending-Type Bernstein\u2013Durrmeyer Operators"],"prefix":"10.3390","volume":"12","author":[{"given":"Yu-Jie","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9199-0502","authenticated-orcid":false,"given":"Wen-Tao","family":"Cheng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6202-8311","authenticated-orcid":false,"given":"Wen-Hui","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pei-Xin","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,21]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Proof of the theorem of Weierstrass based on the calculus of probabilities","volume":"2","author":"Bernstein","year":"1913","journal-title":"Commun. Soc. Math. Kharkow"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1186\/s13660-018-1653-7","article-title":"Approximation properties of \u03bb-Bernstein operators","volume":"2018","author":"Cai","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.jmaa.2016.12.075","article-title":"Approximation of functions by a new family of generalized Bernstein operators","volume":"450","author":"Chen","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kajla, A., Mursaleen, M., and Acar, T. (2020). Durrmeyer-type generalization of parametric Bernstein operators. Symmetry, 12.","DOI":"10.3390\/sym12071141"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s43036-021-00172-z","article-title":"Blending type approximation by modified Bernstein operators","volume":"7","author":"Acu","year":"2022","journal-title":"Adv. Oper. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.2989\/16073606.2019.1639843","article-title":"Blending type approximation by bivariate generalized Bernstein type operators","volume":"43","author":"Baxhaku","year":"2020","journal-title":"Quaest. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"9407","DOI":"10.1002\/mma.7368","article-title":"Blending-type approximation by Lupa\u015f\u2013Durrmeyer-type operators involving P\u00f3lya distribution","volume":"44","author":"Kajla","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"349","DOI":"10.2298\/FIL2201349K","article-title":"Durrmeyer-type generalization of \u03bc-Bernstein operators","volume":"36","author":"Kajla","year":"2022","journal-title":"Filomat"},{"key":"ref_9","first-page":"975","article-title":"Approximation of functions by genuine Bernstein-Durrmeyer type operators","volume":"12","author":"Acat","year":"2018","journal-title":"J. Math. Inequal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1007\/s11766-020-3918-y","article-title":"Approxiamtion properties for the genuine modified Bernstein-Durrmeyer-Stancu operators","volume":"35","author":"Cai","year":"2020","journal-title":"Appl. Math. J. Chin. Univ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1007\/s40995-020-00919-y","article-title":"On new modification of Bernstein operators: Theory and applications","volume":"44","author":"Usta","year":"2020","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_12","first-page":"6694032","article-title":"Approximation theorem for new modification of q-Bernstein operators on (0, 1)","volume":"2020","author":"Wu","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1831","DOI":"10.3934\/math.2022105","article-title":"Some approximation results for the new modification of Bernstein-Beta operators","volume":"7","author":"Cai","year":"2022","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kajla, A., and Micl\u01ceu\u015f, D. (2022). Modified Bernstein-Durrmeyer type operators. Mathematics, 10.","DOI":"10.3390\/math10111876"},{"key":"ref_15","first-page":"1699","article-title":"Parametric generalization of the modified Bernstein operators","volume":"63","author":"Kanat","year":"2022","journal-title":"Filomat"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"DeVore, R.A., and Lorentz, G.G. (1993). Constructive Approximation, Springer.","DOI":"10.1007\/978-3-662-02888-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/1385-7258(88)90007-8","article-title":"On Lipschitz type maximal functions and their smoothness spaces","volume":"90","author":"Lenze","year":"1988","journal-title":"Indag. Math. (Proc.)"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"173","DOI":"10.2298\/FIL1301173O","article-title":"Local approximation for certain King-type operator","volume":"27","year":"2013","journal-title":"Filomat"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ditzion, D., and Totik, V. (1987). Moduli of Smoothness, Springer.","DOI":"10.1007\/978-1-4612-4778-4"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:47:22Z","timestamp":1760147242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,21]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010005"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010005","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,21]]}}}