Bernstein-Schoenberg operator with knots at the q-integers
MATHEMATICAL AND COMPUTER MODELLING, vol.56, pp.56-59, 2012 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 56
- Publication Date: 2012
- Doi Number: 10.1016/j.mcm.2011.12.049
- Journal Name: MATHEMATICAL AND COMPUTER MODELLING
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.56-59
- Keywords: Bernstein-Schoenberg operator, B-spline, Marsden's identity, q-integer
- Dokuz Eylül University Affiliated: Yes
Abstract
We consider a special knot sequence u(i+1) = qu(i) + 1 and define a one parameter family of Bernstein-Schoenberg operators. We prove that this operator converges to f uniformly for all f in C[0, 1]. This operator also inherits the geometric properties of the classical Bernstein-Schoenberg operator. Moreover we show that the error function E-m,E-n has a particular symmetry property, that is E-m,E-n(f; x; q) = E-m,E-n(f; 1 - x, 1/q) provided that f is symmetric on [0, 1]. (C) 2012 Elsevier Ltd. All rights reserved.