Bernstein–Schoenberg operator with knots at the qq-integers

We consider a special knot sequence ui+1=qui+1ui+1=qui+1 and define a one parameter family of Bernstein–Schoenberg operators. We prove that this operator converges to ff uniformly for all ff in C[0,1]C[0,1]. This operator also inherits the geometric properties of the classical Bernstein–Schoenberg operator. Moreover we show that the error function Em,nEm,n has a particular symmetry property, that is Em,n(f;x;q)=Em,n(f;1−x,1/q)Em,n(f;x;q)=Em,n(f;1−x,1/q) provided that ff is symmetric on [0,1][0,1].