Abstract
The notion of a saturation class, due to Favard, can be explained as follows. (For a more general discussion and for the determination of the saturation classes for trigonometric approximation, compare the article of Sunouchi [4].) We try to approximate continuous functions f (x) by a given sequence of linear operators L n (f, x). Sometimes the following phenomena occur. There is a subclass K of the space C of all continuous functions for which the degree of approximation of f by L n (f) is optimal. A still better approximation happens only for functions f of a very narrow subclass K o of K, for which L n (f) = f. In this situation, K is called the saturation class of the operators L n .
This work has been in part supported by the Contract no. AFOSR 424–63 of the Office of Scientific Research, United States Air Force. Syracuse University
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References
de Leeuw, K. On the degree of approximation by Bernstein polynomials Journ. d’Analyse Math. 7 (1959), 89–104
Lorentz, G.G. Bernstein polynomials University of Toronto Press, Toronto, 1953
Lorentz, G. G. The degree of approximation by polynomials with positive coefficients. Math. Annalen 151 (1963) 239–251
Sunouchi, G. Saturation in the theory of best approximation These Proceedings
Widder, D.V. The Laplace Transform Princeton University Press, 1946
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Lorentz, G.G. (1964). Inequalities and the Saturation Classes of Bernstein Polynomials. In: Butzer, P.L., Korevaar, J. (eds) On Approximation Theory / Über Approximationstheorie. ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Nummerischen Mathematik / Série Internationale D’Analyse Numérique, vol 5 . Springer, Basel. https://doi.org/10.1007/978-3-0348-4131-3_19
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DOI: https://doi.org/10.1007/978-3-0348-4131-3_19
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