A sandwich-iterative numerical algorithm for improved approximation by Bernstein’s operator using statistical perspectives

by Ashok Sahai and Shanaz Ansari Wahid.

Abstract: The celebrated Weierstrass Approximation Theorem (1885) heralded intermittent interest in polynomial approximation, which continues undiminished even as of today. In 1912, the great mathematician Bernstein provided an interesting proof of this theorem, as also displayed a sequence of polynomials that would approximate a given function f ? C [0, 1]. This paper aims at constructing a sandwich-iterative computerizable numerical algorithm for an improved approximation by ‘Bernstein’s Polynomial’. The iterative algorithm uses the ‘statistical perspectives’ for using the information aposteriori about the unknown function ‘f’ available in terms of its known values at the ‘equidistant-knots in C [0, 1] rather more fully. Any typical iteration uses the twin statistical concepts of ‘Minimum Mean Square Error (MMSE)’, and ‘Unbiasedness’; the latter concept being sandwiched by the former. The potential of the achievable improvements through the proposed ‘computerizable numerical iterative algorithm is tried to be brought out/ illustrated per an ‘empirical study’ for which the function ‘f’ is assumed to be known in the sense of simulation.

Key Words: Bernstein operator; simulated empirical study

Ashok Sahai, sahai.ashok@gmail.com
Shanaz Ansari Wahid, shanazw@hotmail.com

Editor: Ahmed H. Youssef, ahyoussef@hotmail.com

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