Approximation Properties of New Lupas-type Operators

In this book I studied the approximation properties of a sequence of Lupas-type operators L_n (f;x) on the space C[0∞). Then I define the sequence L ̃_n (f;x) which represented a generalization of the operators L_n (f;x) on the space C_ρ [0∞). I applying the development of Bauer and Donner on Korovkin theorem and define the m-th order moment T ̃_(nm) (x) also I proved the Voronoviskaja-type asymptotic formula for the operators L ̃_n.Finally I define the sequence L ̃_(nm) (f;x) on the space C_(ρq) ([0∞)×[0∞)) which represented an extension of the operators L ̃_n on two dimensions (xy) also applying the development of Bauer and Donner on Korovkin theorem then proved the Voronoviskaja-type asymptotic formula for the operators L ̃_(nm) (f;xy). All above results lead us to show that the three operators L_n L ̃_n and L ̃_(nm) have the same order of approximation.