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Date: 15-5-2019
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Date: 1-10-2019
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The polynomials defined by
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(1) |
where is a binomial coefficient. The Bernstein polynomials of degree
form a basis for the power polynomials of degree
. The first few polynomials are
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(2) |
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
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(10) |
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(11) |
The Bernstein polynomials are implemented in the Wolfram Language as BernsteinBasis[n, i, t].
The Bernstein polynomials have a number of useful properties (Farin 1993). They satisfy symmetry
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(12) |
positivity
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(13) |
for , normalization
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(14) |
and with
has a single unique local maximum of
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(15) |
occurring at .
The envelope of the Bernstein polynomials
for
, 1, ...,
(Mabry 2003) is given by
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(16) |
illustrated above for .
REFERENCES:
Bernstein, S. "Démonstration du théorème de Weierstrass fondée sur le calcul des probabilities." Comm. Soc. Math. Kharkov 13, 1-2, 1912.
Farin, G. Curves and Surfaces for Computer Aided Geometric Design. San Diego: Academic Press, 1993.
Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 222, 1971.
Kac, M. "Une remarque sur les polynomes de M. S. Bernstein." Studia Math. 7, 49-51, 1938.
Kac, M. "Reconnaissance de priorité relative à ma note, 'Une remarque sur les polynomes de M. S. Bernstein.' " Studia Math. 8, 170, 1939.
Lorentz, G. G. Bernstein Polynomials. Toronto: University of Toronto Press, 1953.
Mabry, R. "Problem 10990." Amer. Math. Monthly 110, 59, 2003.
Mathé, P. "Approximation of Hölder Continuous Functions by Bernstein Polynomials." Amer. Math. Monthly 106, 568-574, 1999.
Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, p. 101, 1941.
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دراسة: عدم ترتيب الغرفة قد يدل على مشاكل نفسية
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علماء: تغير المناخ تسبب في ارتفاع الحرارة خلال موسم الحج
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جهود مكثفة وطباعة عشرات الآلاف من المنشورات .. استعدادات العتبة العلوية المقدسة لعيد الغدير الأغر
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