Modified MKZ Operators Providing Better Estimation on Chosen Part of [1/2, 1)

by Shanaz Ansari Wahid, Ashok Sahai & Vrijesh Tripathi.

Abstract: Recently, özarslan & Duman (2007) introduced a modification of the Meyer-Konig and Zeller (MKZ) operators, (Say MMKZO1k,n (f; x)) which preserves the test-functions f0(x) = 1 and f2(x) = x2, and showed that it provides a better estimation than the classical MKZ-operators on the interval [1/2, 1) with respect to the modulus of continuity and the Lipschitz class functions. In the present paper, we provide a more efficient modification. Our motivation is two-fold. Firstly, both of the aforementioned operators are practically numerically impossible, inasmuch as they require the values of the impugned function at infinitely many knots: namely the values of the function “f” we need are f (k/ (k+n)); k = 0, 1, 2, ..., ?. This impracticality is remedied by our present modification of MKZ-operators (Say FVMMKZO2k,n (f; x)) needing only the values of the function “f” we need are f (k/ (k+n)); k = 0, 1, 2, ..., m for providing an efficient estimation in interval [1/2, m/ (m+n)].In fact, in contrast to the ‘Mathematicians’, who are fond of “abstractions”, the ‘Statisticians-and-Computer-Scientists’ are rather more keen to ensure the “applicational/computational efficiency” of any algorithm. We have, therefore, modified impugned operators: Mk,n (f; x) & its modification MMKZO1k,n (f; x) by their “Finite-Versions” proposed by us, namely by “FVMk,n (f; x)” & “FVMMKZO1k,n (f; x)”, respectively. For illustration purpose, we have chosen the interval to be [1/2, 16/21]. Secondly, we found that our modification would be numerically more efficient than özarslan & Duman (2007)'s. To facilitate an illustration of relative efficiency of approximation of our version of ‘Modified MKZ Operators FVMMKZO2k,n (f; x)’ vs. that of ‘özarslan & Duman (2007)’s Operators FVMMKZO1k,n (f; x)’, we have accommodated a brief numerical study in the last section of this paper. In this 'simulated numerical study' we have chosen four illustrative example-functions: namely exp (x), sin (2+x), ln (2+x), and 10x. For the simplicity of the numerical illustration, we have chosen our illustrative m-n-values as m = 16 & n = 5; m = 32 & n = 10; m = 48 & n = 15, assuming that our chosen interval of interest-of-estimation happens to be [1/2, 16/21]. This illustrative numerical-comparison of “Finite-Versions” has amply supported the fact that our modification performs much better, inasmuch as the absolute approximation error for our version of ‘Modified MKZ Operators FVMMKZO2k,n (f; x))’ is significantly less than that for the classical MKZ operators FVMk,n (f; x)), as also than that for the Ozarslan & Duman (2007)'s modification of the Meyer-Konig and Zeller (MKZ) operators, FVMMKZO1k,n (f; x)).

Key Words: Positive Linear Operators, Rate of Convergence, Modulus of Continuity, Lipschitz Class, Numerical Study

Shanaz Ansari Wahid,
Ashok Sahai,
Vrijesh Tripathi,

Editor: : Ahmed H. Youssef,

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