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Linear Combinations of BernsteinPolynomials

Published online by Cambridge University Press:  20 November 2018

P. L. Butzer
P. L. Butzer*
Affiliation:
The University of Toronto and McGill University
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If f(x) is denned on [0, 1], then its corresponding Bernstein polynomial

approaches f(x) uniformly on [0, 1], if f(x) is continuous on [0, 1]. If f(x) is bounded on [0, 1], then at every point x where the second derivative exists (Voronowskaja [7], see also [5])

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 559 - 567
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Bernstein, S. N., Complétement a Varticle de E. Voronowskaja, C. R. Acad. Sci. U.R.S.S. (1932), 86–92.Google Scholar
2. Butzer, P. L., On Bernstein polynomials, Thesis, University of Toronto Library (1951).Google Scholar
3. Feller, W., An introduction to probability theory and its applications (New York, 1950).Google Scholar
4. Jackson, D., The theory of approximation (Amer. Math. Soc. Coll. Publ., vol. 11, New York, 1930).Google Scholar
5. Lorentz, G. G., Bernstein polynomials (Toronto, 1953).Google Scholar
6. Popoviciu, T., Sur Vapproximation des fonctions convexes d'ordre supérieur, Mathematica, Cluj, 10 (1935), 49–54.Google Scholar
7. Voronowskaja, E., Détermination de la forme asymptotique d'approximation des fonctions par les polynomes de M. Bernstein, C. R. Acad. Sci. U.R.S.S. (1932), 79–85.Google Scholar