{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T04:03:52Z","timestamp":1783829032503,"version":"3.55.0"},"reference-count":8,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2009,12,9]],"date-time":"2009-12-09T00:00:00Z","timestamp":1260316800000},"content-version":"vor","delay-in-days":342,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>Starting from the study of the <jats:italic>Shepard nonlinear operator of max-prod\ntype<\/jats:italic> by Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324\u2013326, the <jats:italic>Bernstein max-prod-type operator<\/jats:italic> is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form  (with an unexplicit absolute constant <jats:italic>C<\/jats:italic> &gt; 0) and the\nquestion of improving the order of approximation  is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of  and it can easily be extended to other max\u2010prod operators of Bernstein type. However, for subclasses of functions <jats:italic>f<\/jats:italic> including, for example, that of concave functions, we find the order of approximation <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>f<\/jats:italic>; 1\/<jats:italic>n<\/jats:italic>), which for many functions <jats:italic>f<\/jats:italic> is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape\u2010preserving properties are obtained.<\/jats:p>","DOI":"10.1155\/2009\/590589","type":"journal-article","created":{"date-parts":[[2009,12,10]],"date-time":"2009-12-10T07:07:07Z","timestamp":1260428827000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["Approximation and Shape Preserving Properties of the Bernstein Operator of Max\u2010Product Kind"],"prefix":"10.1155","volume":"2009","author":[{"given":"Barnab\u00e1s","family":"Bede","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lucian","family":"Coroianu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sorin G.","family":"Gal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2009,12,9]]},"reference":[{"key":"e_1_2_7_1_2","doi-asserted-by":"crossref","first-page":"494","DOI":"10.20965\/jaciii.2006.p0494","article-title":"Max-product Shepard approximation operators","volume":"10","author":"Bede B.","year":"2006","journal-title":"Journal of Advanced Computational Intelligence and Intelligent Informatics"},{"key":"e_1_2_7_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2007.11.007"},{"key":"e_1_2_7_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4703-2"},{"key":"e_1_2_7_4_2","unstructured":"BedeB.andGalS. G. Approximation by nonlinear Bernstein and Favard-Sz\u00e1sz Mirakjan operators of max-product kind to appear inJournal of Concrete and Applicable Mathematics."},{"key":"e_1_2_7_5_2","first-page":"45","article-title":"Deux remarques sur les fonctions convexes","volume":"220","author":"Popoviciu T.","year":"1938","journal-title":"Bull. Soc. Sci. Acad. Roumaine"},{"key":"e_1_2_7_6_2","first-page":"195","article-title":"The preservation of the property of the quasiconvexity of higher order by Bernstein polynomials","volume":"25","author":"P\u0103lt\u0103nea R.","year":"1996","journal-title":"Revue d\u2032Analyse Num\u00e9rique et de Th\u00e9orie de l\u2032Approximation"},{"key":"e_1_2_7_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF03323413"},{"key":"e_1_2_7_8_2","doi-asserted-by":"publisher","DOI":"10.2307\/2375007"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2009\/590589.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2009\/590589.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2009\/590589","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T14:57:09Z","timestamp":1719068229000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2009\/590589"}},"subtitle":[],"editor":[{"given":"Narendra Kumar","family":"Govil","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1155\/2009\/590589"],"URL":"https:\/\/doi.org\/10.1155\/2009\/590589","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]},"assertion":[{"value":"2009-08-31","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-10-25","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-12-09","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"590589"}}