Using the $q$-Bernstein basis, we construct a new sequence $\{ L_{n} \}$ of positive linear operators in $C[0,1].$ We study its approximation properties and the rate of convergence in terms of modulus of continuity.
Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q<1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.