{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T19:09:03Z","timestamp":1761419343078,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,10,10]],"date-time":"2019-10-10T00:00:00Z","timestamp":1570665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Sz\u00e1sz-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Sz\u00e1sz-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Sz\u00e1sz-Mirakjan Beta-type operators.<\/jats:p>","DOI":"10.3390\/axioms8040111","type":"journal-article","created":{"date-parts":[[2019,10,11]],"date-time":"2019-10-11T03:07:11Z","timestamp":1570763231000},"page":"111","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Approximation Properties of an Extended Family of the Sz\u00e1sz\u2013Mirakjan Beta-Type Operators"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","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"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"}]},{"given":"G\u00fcrhan","family":"\u0130\u00e7\u00f6z","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Gazi University, Ankara TR-06500, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5363-2453","authenticated-orcid":false,"given":"Bayram","family":"\u00c7ekim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Gazi University, Ankara TR-06500, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"239","DOI":"10.6028\/jres.045.024","article-title":"Generalizations of S. Bernstein\u2019s polynomials to the infinite interval","volume":"45","year":"1950","journal-title":"J. Res. Nat. Bur. Stand."},{"key":"ref_2","first-page":"201","article-title":"Approximation des fonctions continues au moyen de polyn\u00f4mes de la forme e\u2212nx\u2211k=0mnck,nxk","volume":"31","author":"Mirakjan","year":"1941","journal-title":"C. R. (Doklady) Acad. Sci. URSS (New Ser.)"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1016\/j.aml.2006.10.007","article-title":"Sz\u00e1sz-Mirakjan-type operators providing a better error estimation","volume":"20","author":"Duman","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"ref_4","first-page":"77","article-title":"Approximation with an arbitrary order by generalized Sz\u00e1sz-Mirakjan operators","volume":"59","author":"Gal","year":"2014","journal-title":"Stud. Univ. Babe\u015f-Bolyai Math."},{"key":"ref_5","first-page":"41","article-title":"Approximation properties of the q-Sz\u00e1sz-Mirakjan-Beta operators","volume":"3","author":"Gupta","year":"2012","journal-title":"Indian J. Ind. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2005.07.036","article-title":"Convergence of derivatives for certain mixed Sz\u00e1sz-Beta operators","volume":"321","author":"Gupta","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"373","DOI":"10.4208\/ata.2013.v29.n4.6","article-title":"Approximation properties by q-Durrmeyer-Stancu operators","volume":"29","author":"Mohapatra","year":"2013","journal-title":"Anal. Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"575","DOI":"10.29020\/nybg.ejpam.v11i3.3314","article-title":"A general family of the Srivastava-Gupta operators preserving linear functions","volume":"11","author":"Gupta","year":"2018","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_9","first-page":"1","article-title":"On simultaneous approximation by Sz\u00e1sz-beta operators","volume":"21","author":"Gupta","year":"1995","journal-title":"Soochow J. Math."},{"key":"ref_10","first-page":"378","article-title":"Modified Sz\u00e1sz-Mirakjan operators of integral form","volume":"24","year":"2008","journal-title":"Carpathian J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5437","DOI":"10.1002\/mma.4397","article-title":"Some approximation results involving the q-Sz\u00e1sz-Mirakjan-Kantorovich type operators via Dunkl\u2019s generalization","volume":"40","author":"Srivastava","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., \u00d6zger, F., and Mohiuddine, S.A. (2019). Construction of Stancu-type Bernstein operators based on B\u00e9zier bases with shape parameter \u03bb. Symmetry, 11.","DOI":"10.3390\/sym11030316"},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.jat.2007.11.003","article-title":"Approximation theorems for localized Sz\u00e1sz-Mirakjan operators","volume":"152","author":"Xie","year":"2008","journal-title":"J. Approx. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1007\/s11253-012-0614-4","article-title":"Approximation of absolutely continuous functions by Stancu Beta operators","volume":"63","author":"Zeng","year":"2012","journal-title":"Ukrainian Math. J."},{"key":"ref_16","first-page":"53","article-title":"Better error estimates for Sz\u00e1sz-Mirakjan-Beta operators","volume":"10","author":"Duman","year":"2008","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1785","DOI":"10.1016\/j.aml.2011.04.032","article-title":"A-Statistical approximation of generalized Sz\u00e1sz-Mirakjan-Beta operators","volume":"24","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Qi, Q.-L., and Zhang, Y.-P. (2009). Pointwise approximation for certain mixed Sz\u00e1sz-Beta operators. Further Progress in Analysis, Proceedings of the Sixth International Conference (ISAAC 2002) on Clifford Algebras and Their Applications in Mathematical Physics, Tennessee Technological University, Cookeville, TN, USA, 20\u201325 May 2002, World Scientific Publishing Company.","DOI":"10.1142\/9789812837332_0009"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0377-0427(94)90187-2","article-title":"Generalized incomplete gamma functions with applications","volume":"55","author":"Chaudhry","year":"1994","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0377-0427(96)00102-1","article-title":"Extension of Euler\u2019s beta function","volume":"78","author":"Chaudhry","year":"1997","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4601","DOI":"10.1016\/j.cam.2010.04.019","article-title":"Extension of gamma, beta and hypergeometric functions","volume":"235","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","first-page":"1731","article-title":"Some classes of generating relations associated with a family of the generalized Gauss type hypergeometric functions","volume":"9","author":"Lin","year":"2015","journal-title":"Appl. Math. Inform. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"238","DOI":"10.3390\/axioms1030238","article-title":"A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions","volume":"1","author":"Srivastava","year":"2012","journal-title":"Axioms"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"DeVore, R.A., and Lorentz, G.G. (1993). Constructive Approximation, Springer.","DOI":"10.1007\/978-3-662-02888-9"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/8\/4\/111\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:28:55Z","timestamp":1760189335000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/8\/4\/111"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,10]]},"references-count":24,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["axioms8040111"],"URL":"https:\/\/doi.org\/10.3390\/axioms8040111","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2019,10,10]]}}}