Geodesic
King type modification of $q$-Bernstein-Schurer operators
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 805-817
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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.
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.
DOI : 10.1007/s10587-013-0054-9
Classification : 41A10, 41A25, 41A36
Keywords: King type operator; $q$-Bernstein-Schurer operator; Korovich type approximation theorem; rate of convergence; Voronovskaja-type result; modulus of continuity 
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Ren, Mei-Ying; Zeng, Xiao-Ming. King type modification of $q$-Bernstein-Schurer operators. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 805-817. doi: 10.1007/s10587-013-0054-9

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