King type modification of $q$-Bernstein-Schurer operators

summary:Very recently the $q$-Bernstein-Schurer operators which reproduce only constant function were introduced and studied by C. V. Muraru (2011). Inspired by J. P. King, Positive linear operators which preserve $x^{2}$ (2003), in this paper we modify $q$-Bernstein-Schurer operators to King type modification of $q$-Bernstein-Schurer operators, so that these operators reproduce constant as well as quadratic test functions $x^{2}$ and study the approximation properties of these operators. We establish a convergence theorem of Korovkin type. We also get some estimations for the rate of convergence of these operators by using modulus of continuity. Furthermore, we give a Voronovskaja-type asymptotic formula for these operators