voronovskaya theorem
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2016 ◽  
Vol 53 (3) ◽  
pp. 379-398 ◽  
Author(s):  
Tuncer Acar ◽  
Ali Aral ◽  
Ioan Raşa

In this paper, we study the k-th order Kantorovich type modication of Szász—Mirakyan operators. We first establish explicit formulas giving the images of monomials and the moments up to order six. Using this modification, we present a quantitative Voronovskaya theorem for differentiated Szász—Mirakyan operators in weighted spaces. The approximation properties such as rate of convergence and simultaneous approximation by the new constructions are also obtained.


1986 ◽  
Vol 38 (5) ◽  
pp. 1081-1093 ◽  
Author(s):  
M. J. Marsden

In [7] Schoenberg introduced the following variation-diminishing spline approximation methods.Let m > 1 be an integer and let Δ = {xi} be a biinfinite sequence of real numbers with xi ≧ xi + l < xi+m. To a function f associate the spline function Vf of order m with knots Δ defined by(1.1)whereand the Nj(x) are B-splines with support xj < x < xj+m normalized so that ΣjNj(x) = 1. See, e.g., [2] for a precise definition of the Nj(x) and a discussion of the properties of Vf.


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