The celebrated Weierstrass Approximation Theorem (1885) heralded intermittent interest in polynomial approximation, which continues unabated even as of today. The great Russian mathematician Bernstein, in 1912, not only provided an interesting proof of the Weierstrass’ theorem, but also displayed a sequence of the polynomials which approximate the given function . An efficient ‘Combinatorial-Probabilistic Dual-Fusion’ version of the modification of Bernstein’s Polynomial Operator is proposed. The potential of the aforesaid improvement is tried to be brought forth and illustrated through an empirical study, for which the function is assumed to be known in the sense of simulation.
K. Weierstrass, “Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen Veranderlichen Sitzungsberichteder,” Koniglich Preussischen Akademie der Wissenschcaften zu Berlin, 1885, pp. 633-639 & pp. 789-805.
A. Sahai, “An Iterative Algorithm for Improved Approximation by Bernstein’s Operator Using Statistical Perspective,” Applied Mathematics and Computation, Vol. 149, No. 2, 2004, pp. 327-335.
A. Sahai and G. Prasad, “Sharp Estimates of Approximation by Some Positive Linear Operators,” Bulletin of the Australian Mathematical Society, Vol. 29, No. 1, 1984, pp. 13-18. doi:10.1017/S0004972700021225
S. A. Wahid, A. Sahai and M. R. Acharya, “A Computerizable Iterative-Algorithmic Quadrature Operator Using an Efficient Two-Phase Modification of Bernstein Polynomial,” International Journal of Engineering and Technology, Vol. 1, No. 3, 2009, pp. 104-108.