{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T01:29:26Z","timestamp":1775093366294,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,2]],"date-time":"2019-03-02T00:00:00Z","timestamp":1551484800000},"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>We construct Stancu-type Bernstein operators based on B\u00e9zier bases with shape parameter     \u03bb \u2208 [ \u2212 1 , 1 ]     and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type    \u03bb   -Bernstein operators and study their approximation behaviors.<\/jats:p>","DOI":"10.3390\/sym11030316","type":"journal-article","created":{"date-parts":[[2019,3,4]],"date-time":"2019-03-04T05:45:36Z","timestamp":1551678336000},"page":"316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":91,"title":["Construction of Stancu-Type Bernstein Operators Based on B\u00e9zier Bases with Shape Parameter \u03bb"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4135-2091","authenticated-orcid":false,"given":"Faruk","family":"\u00d6zger","sequence":"additional","affiliation":[{"name":"Department of Engineering Sciences, \u0130zmir Katip \u00c7elebi University, \u0130zmir 35620, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9050-9104","authenticated-orcid":false,"given":"S. A.","family":"Mohiuddine","sequence":"additional","affiliation":[{"name":"Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,2]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"D\u00e9monstration du th\u00e9or\u00e8me de Weierstrass fond\u00e9e sur le calcul des probabilit\u00e9s","volume":"13","author":"Bernstein","year":"1913","journal-title":"Comm. Soc. Math. Kharkow"},{"key":"ref_2","first-page":"31","article-title":"Asupra unei generalizari a polinoamelor lui Bernstein","volume":"14","author":"Stancu","year":"1969","journal-title":"Studia Univ. Babes-Bolyai Ser. Math.-Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1007\/s11785-016-0633-5","article-title":"Approximation by (p,q)-Baskakov-Durrmeyer-Stancu operators","volume":"12","author":"Acar","year":"2018","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.amc.2017.02.007","article-title":"Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators","volume":"306","author":"Baxhaku","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1186\/s13660-017-1298-y","article-title":"A new kind of Bernstein-Schurer-Stancu-Kantorovich-type operators based on q-integers","volume":"2017","author":"Chauhan","year":"2017","journal-title":"J. Inequal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1016\/j.amc.2015.03.135","article-title":"Some approximation results by (p,q)-analogue of Bernstein-Stancu operators","volume":"264","author":"Mursaleen","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_7","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_8","unstructured":"Ye, Z., Long, X., and Zeng, X.-M. (2010, January 24\u201327). Adjustment algorithms for B\u00e9zier curve and surface. Proceedings of the International Conference on Computer Science and Education, Hefei, China."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1186\/s13660-018-1688-9","article-title":"The B\u00e9zier variant of Kantorovich type \u03bb-Bernstein operators","volume":"2018","author":"Cai","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1186\/s13660-018-1795-7","article-title":"Approximation properties of \u03bb-Kantorovich operators","volume":"2018","author":"Acu","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_11","unstructured":"\u00d6zger, F. (arXiv, 2018). Some general statistical approximation results for \u03bb-Bernstein operators, arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1080\/01630563.2014.970646","article-title":"On pointwise convergence of q-Bernstein operators and their q-derivatives","volume":"36","author":"Acar","year":"2015","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1007\/s40995-017-0154-8","article-title":"On Kantorovich modification of (p,q)-Bernstein operators","volume":"42","author":"Acar","year":"2018","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1007\/s40995-016-0045-4","article-title":"Approximation by bivariate (p,q)-Bernstein-Kantorovich operators","volume":"42","author":"Acar","year":"2018","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1186\/s13660-016-1045-9","article-title":"On Kantorovich modification of (p,q)-Baskakov operators","volume":"2016","author":"Acar","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s00025-015-0441-7","article-title":"Approximation properties of bivariate extension of q-Bernstein-Schurer-Kantorovich operators","volume":"67","author":"Acu","year":"2015","journal-title":"Results Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.amc.2014.05.134","article-title":"On generalized integral Bernstein operators based on q-integers","volume":"242","author":"Mishra","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"7749","DOI":"10.1002\/mma.4559","article-title":"Construction of a new family of Bernstein-Kantorovich operators","volume":"40","author":"Mohiuddine","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"961","DOI":"10.7153\/jmi-2018-12-73","article-title":"Durrmeyer type (p,q)-Baskakov operators preserving linear functions","volume":"12","author":"Mohiuddine","year":"2018","journal-title":"J. Math. Inequal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1186\/s13660-018-1693-z","article-title":"Genuine modified Bernstein-Durrmeyer operators","volume":"2018","author":"Mohiuddine","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1016\/j.amc.2015.04.090","article-title":"On (p,q)-analogue of Bernstein operators","volume":"266","author":"Mursaleen","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.amc.2013.11.095","article-title":"A Korovkin\u2019s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vall\u00e9e Poussin mean","volume":"228","author":"Braha","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_23","first-page":"283","article-title":"Approximation by means of the Sz\u00e1sz-B\u00e9zier integral operators","volume":"14","author":"Srivastava","year":"2004","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ditzian, Z., and Totik, V. (1987). Moduli of Smoothness, Springer.","DOI":"10.1007\/978-1-4612-4778-4"},{"key":"ref_25","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_26","unstructured":"Peetre, J. (1963). A Theory of Interpolation of Normed Spaces, Notas Mat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"173","DOI":"10.2298\/FIL1301173O","article-title":"Local approximation for certain King type operators","volume":"27","author":"Ozarslan","year":"2013","journal-title":"Filomat"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.amc.2003.07.023","article-title":"The properties of generalized Bernstein polynomials of two variables","volume":"156","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/0021-9045(89)90095-6","article-title":"Some properties of two-demansional Bernstein polynomials","volume":"59","author":"Martinez","year":"1989","journal-title":"J. Approx. Theory"},{"key":"ref_30","first-page":"17","article-title":"On the convergence of linear positive operators in the space of continuous functions of two variables","volume":"115","author":"Volkov","year":"1957","journal-title":"Doklakad Nauk SSSR"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/316\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:35:51Z","timestamp":1760186151000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,2]]},"references-count":30,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["sym11030316"],"URL":"https:\/\/doi.org\/10.3390\/sym11030316","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,2]]}}}