{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T04:45:39Z","timestamp":1768193139677,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,8]],"date-time":"2020-07-08T00:00:00Z","timestamp":1594166400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation are established. Finally, the theoretical results are demonstrated by using MAPLE software.<\/jats:p>","DOI":"10.3390\/sym12071141","type":"journal-article","created":{"date-parts":[[2020,7,10]],"date-time":"2020-07-10T09:45:31Z","timestamp":1594374331000},"page":"1141","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Durrmeyer-Type Generalization of Parametric Bernstein Operators"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4273-4830","authenticated-orcid":false,"given":"Arun","family":"Kajla","sequence":"first","affiliation":[{"name":"School of Basic Sciences, Faculty of Mathematics, Central University of Haryana, Haryana-123029, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4128-0427","authenticated-orcid":false,"given":"Mohammad","family":"Mursaleen","sequence":"additional","affiliation":[{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"},{"name":"Department of Computer Science and Information Engineering, Asia University, Taichung 41354, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0982-9459","authenticated-orcid":false,"given":"Tuncer","family":"Acar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Selcuk University, Selcuklu, Konya 42003, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,8]]},"reference":[{"key":"ref_1","first-page":"789","article-title":"\u00dcber die analytische Darstellbarkeit sogennanter willk\u00fcrlicher Functionen einer reellen Ver\u00e4nderlichen","volume":"633\u2013639","author":"Weierstrass","year":"1885","journal-title":"Sitzungsber. 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Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s40574-015-0034-0","article-title":"q-Bernstein-Schurer-Kantorovich type operators","volume":"8","author":"Agrawal","year":"2015","journal-title":"Boll. Unione Mat. Ital."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1007\/s13398-018-0525-9","article-title":"Approximation by Jakimovski-Leviatan operators of Durrmeyer type involving multiple Appell polynomials","volume":"113","author":"Ansari","year":"2019","journal-title":"Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM"},{"key":"ref_11","first-page":"98","article-title":"Positive linear operators preserving \u03c4 and \u03c42","volume":"2","author":"Acar","year":"2019","journal-title":"Constr. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.2298\/FIL1906517M","article-title":"Approximation by Jakimovski-Leviatan-Stancu-Durrmeyer type operators","volume":"33","author":"Mursaleen","year":"2019","journal-title":"Filomat"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Acu, A.M., and Kajla, A. (2020). Blending Type Approximation by Generalized Bernstein Operators, submitted for publication.","DOI":"10.1007\/s43036-021-00172-z"},{"key":"ref_14","first-page":"961","article-title":"On convergence of linear positive operators in the space of continuous functions","volume":"90","author":"Korovkin","year":"1959","journal-title":"Dokl. Akad. Nouk. SSR"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s11117-015-0338-4","article-title":"The new forms of Voronovskaya\u2019s theorem in weighted spaces","volume":"20","author":"Acar","year":"2016","journal-title":"Positivity"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1515\/gmj-2016-0007","article-title":"Quantitative q-Voronovskaya and q-Gr\u00fcss-Voronovskaya-type results for q-Szasz operators","volume":"23","author":"Acar","year":"2016","journal-title":"Georgian Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.1080\/01630563.2016.1219743","article-title":"Voronovskaya type theorems in weighted spaces","volume":"37","author":"Erencin","year":"2016","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3391","DOI":"10.1002\/mma.3784","article-title":"q-Voronovskaya type theorems for q-Baskakov operators","volume":"39","author":"Ulusay","year":"2016","journal-title":"Math. Method Appl. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"173","DOI":"10.2298\/FIL1301173O","article-title":"Local approximation for certain King type operators","volume":"27","year":"2013","journal-title":"Filomat"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Devore, R.A., and Lorentz, G.G. (1993). Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Springer.","DOI":"10.1007\/978-3-662-02888-9"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ditzian, Z., and Totik, V. (1987). 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