Date of Award
Doctor of Philosophy
Here, we give the approximation properties with rates of generalized discrete versions of Picard, Gauss-Weierstrass, and Poisson-Cauchy singular operators. We cover both the unitary and non-unitary cases of the operators above. We present quantitatively the point-wise and uniform convergences of these operators to the unit operator by involving the higher modulus of smoothness of a uniformly continuous function. We also establish our results with respect to L_p norm, 1≤p<∞. Additionally, we state asymptotic Voronovskaya type expansions for these operators. Moreover, we study the fractional generalized smooth discrete singular operators on the real line regarding their convergence to the unit operator with fractional rates in the uniform norm. Then, we give our results for the operators mentioned above over the real line regarding their simultaneous global smoothness preservation property with respect to L_p norm for 1≤p≤∞, by involving higher order moduli of smoothness. Here we also obtain Jackson type inequalities of simultaneous approximation which are almost sharp, containing neat constants, and they reflect the high order of differentiability of involved function. Next, we cover the approximation properties of on the general complex-valued discrete singular operators over the real line regarding their convergence to the unit operator with rates in the L_p norm for 1≤p≤∞. Finally, we establish the approximation properties of multivariate generalized discrete versions of these operators over R^N,N≥1. We give pointwise, uniform, and L_p convergence of the operators to the unit operator by involving the multivariate higher order modulus of smoothness.
dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Kester, Merve, "Approximations by Generalized Discrete Singular Operators" (2016). Electronic Theses and Dissertations. 1403.