Bernstein-Schoenberg operator with knots at the q-integers

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.