An Efficient Combinatorial-Probabilistic Dual-Fusion Modification of Bernstein’s Polynomial Approximation Operator

Author(s)
Shanaz Ansari Wahid

ABSTRACT

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.

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.

Cite this paper

nullS. Wahid, "An Efficient Combinatorial-Probabilistic Dual-Fusion Modification of Bernstein’s Polynomial Approximation Operator,"*Applied Mathematics*, Vol. 2 No. 12, 2011, pp. 1535-1538. doi: 10.4236/am.2011.212218.

nullS. Wahid, "An Efficient Combinatorial-Probabilistic Dual-Fusion Modification of Bernstein’s Polynomial Approximation Operator,"

References

[1] W. Cheney and D. Kincaid, “Numerical Mathematics and Computing,” Brooks/Cole Publishing Company, Belmont, 1994.

[2] P. J. Heartley, A. Wynn-Evans, “A Structured Introduction to Numerical Mathematics,” Stanley Thornes, Belmont, 1979.

[3] B. F. Polybon, “Applied Numerical Analysis,” PWS-Kent, Boston, 1992.

[4] A. Sheilds, “Polynomial Approximation,” The Math Intelligencer, Vol. 9, No. 3, 1987, pp. 5-7.

[5] 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.

[6] N. L. Carothers, “A Short Course on Approximation Theory,” Bowling Green State University, Bowling Green, OH, 1998.

[7] E. R. Hedrick, “The Significance of Weirstrass Theorem,” The American Mathematical Monthly, Vol. 20, 1927, pp. 211-213. doi:10.2307/2974105

[8] G. G. Lorentz, “Approximation of Functions,” Chelsea, New York, 1986.

[9] N. L. Carothers, “Real Analysis,” Cambridge University Press, Cambridge, 2000.

[10] 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. doi:10.1016/S0096-3003(03)00081-X

[11] 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

[12] A. Sahai and S. Verma, “Efficient Quadrature Operator Using Dual-Perspectives-Fusion Probabilistic Weights,” International Journal of Engineering and Technology, Vol. 1, No. 1, 2009, pp. 1-8.

[13] 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.

[14] A. Sahai, S. A. Wahid and A. Sinha, “A Positive Linear Operator Using Probabilistic Approach,” Journal of Applied Science, Vol. 6, No. 12, 2006, pp. 2662-2665.

[1] W. Cheney and D. Kincaid, “Numerical Mathematics and Computing,” Brooks/Cole Publishing Company, Belmont, 1994.

[2] P. J. Heartley, A. Wynn-Evans, “A Structured Introduction to Numerical Mathematics,” Stanley Thornes, Belmont, 1979.

[3] B. F. Polybon, “Applied Numerical Analysis,” PWS-Kent, Boston, 1992.

[4] A. Sheilds, “Polynomial Approximation,” The Math Intelligencer, Vol. 9, No. 3, 1987, pp. 5-7.

[5] 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.

[6] N. L. Carothers, “A Short Course on Approximation Theory,” Bowling Green State University, Bowling Green, OH, 1998.

[7] E. R. Hedrick, “The Significance of Weirstrass Theorem,” The American Mathematical Monthly, Vol. 20, 1927, pp. 211-213. doi:10.2307/2974105

[8] G. G. Lorentz, “Approximation of Functions,” Chelsea, New York, 1986.

[9] N. L. Carothers, “Real Analysis,” Cambridge University Press, Cambridge, 2000.

[10] 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. doi:10.1016/S0096-3003(03)00081-X

[11] 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

[12] A. Sahai and S. Verma, “Efficient Quadrature Operator Using Dual-Perspectives-Fusion Probabilistic Weights,” International Journal of Engineering and Technology, Vol. 1, No. 1, 2009, pp. 1-8.

[13] 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.

[14] A. Sahai, S. A. Wahid and A. Sinha, “A Positive Linear Operator Using Probabilistic Approach,” Journal of Applied Science, Vol. 6, No. 12, 2006, pp. 2662-2665.