Babeş-Bolyai University, Romania
Title: From uniform to statistical convergence of some approximation processes
Our talk represents a journey in the complex world of the sequences of linear operators, the essential ingredient being that of positivity. The research involves, among other things, quantitative estimates for the rate of convergence and asymptotic behavior. At first we are concerned by a general class of linear positive approximation processes designed using series. For continuous and bounded functions defined on unbounded interval we give the error of approximation in terms of the modulus of smoothness. The main goal is to identify functions for which these operators provide uniform convergence to the identity operator over unbounded intervals. We also construct and study integral operators of Kantorovich-type which reproduce the affine functions.For this exploration, special cases are highlighted. The second target of the talk is the study of statistical convergence forclasses of linear positive operators. We present the notion of statistical convergence, a statistical property of moduli of smooth-ness and K-functionals, Korovkin-type approximation theorems via statistical convergence.In our approach, Markov operators play a central role.
In 1995 I obtained my PhD Degree in Mathematics. At Babes – Bolyai University, Faculty of Mathematics and Computer Science, I successively held the following academic positions: Lecturer (1991 - 1997), Associate Professor (1997 - 2002), Professor (2002 – presentday). Since2012 I gained managerial experience asDirector of theMathematics Department. My scientific interestsare inthe areas ofOperator Theory (positive operators, semigroups of operators) and Approximation Theory (Korovkin-type approximation theory, positive approximation processes). Considering these fields, I have participated giving talks at international conferences, for example inSwitzerland: Bern (2016), Denmark: Copenhagen (2015), Italy: Acquafredda di Maratea (2009, 2004, 2000) and Mondello (2005), Cyprus: Paphos (2006), Hungary: Debrecen (2004) andVisegrad (1999), USA: Nashville (2003), Flagstaff (2002) and Rochester (1998).At other conferences I was invited speaker,such as: The 11thJointConference on Mathematicsand Computer Science, May 2016, Eger, Hungary; Seminar of Estonian Doctoral School in Mathematics and Statistics, June 2012, University ofTartu, Estonia;International Conference on Recent Developments in Functional Analysis and Approximation Theory,September 2011, Lecce, Italy;International Conference on Mathematical Sciences November 2010, AbantIzzetBaysal University of Bolu, Turkey. Ihave alsoreceived invitations from various universities as visiting professor:The African University of Science and Technology, Abuja, Nigeria (2009 and 2010), Tishreen University, Latakia, Syria (2008), Ankara University, Turkey (2007), Vienna University of Technology,Austria (2006).In the last years under the ERASMUS program I have benefited from scholarships at Burapha University (2016 - Bangsaen, Thailanda), University of Murcia (2014 - Spain), University of Miskolc (2013 - Hungary), University of Tallin (2012 - Estonia). I am Editor-in-ChiefatJournal of Numerical Analysis and Approximation Theory,Editor at StudiaUniversitatis Babes-Bolyai, Mathematica,Editor at Mathematica (Cluj),member ofthe Editorial Board ofOpen Journal of Discrete Mathematics andJournal of Science and Arts. Also, I ama member of theAMS. As aPhD supervisor, 8 students finalized their PhD thesis, under my guidance. Synthesizing my research activity in terms of quantity, I published 13 books, monographs, lecture notes, 28 papers in Web of Science (ISI) Thomson – Reutersindexed journals, 66papers in other international journals or Proceedings. On GoogleScholarI have accumulated over 670 citations, myh-index and i10-index being 15 and 22, respectively. My favorite motto: When you grow to believe that your work is very important, you arevery near to a nervous breakdown. (Bertrand Russel)