Derivatives of Bernstein polynomials and smoothness

Author:
Z. Ditzian

Journal:
Proc. Amer. Math. Soc. **93** (1985), 25-31

MSC:
Primary 41A10

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766520-7

MathSciNet review:
766520

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Abstract: Equivalence relations between the asymptotic behaviour of derivatives of Bernstein polynomials and the smoothness of the function they approximate are given. This is achieved with an a priori condition that the function is of class with some small . The a priori condition is dropped when a similar equivalence relation using the Katorovich operator is proved.

**[1]**Hubert Berens and George G. Lorentz,*Inverse theorems for Bernstein polynomials*, Indiana Univ. Math. J.**21**(1971/72), 693–708. MR**0296579**, https://doi.org/10.1512/iumj.1972.21.21054**[2]**M. Becker,*An elementary proof of the inverse theorem for Bernšteĭn polynomials*, Aequationes Math.**19**(1979), no. 2-3, 145–150. MR**556718**, https://doi.org/10.1007/BF02189862**[3]**Z. Ditzian,*A global inverse theorem for combinations of Bernšteĭn polynomials*, J. Approx. Theory**26**(1979), no. 3, 277–292. MR**551679**, https://doi.org/10.1016/0021-9045(79)90065-0**[4]**-,*Interpolation and the rate of convergence of Bernstein polynomials*, Approximation Theory. III (W. Cheney, ed.), Academic Press, New York, 1980, pp. 241-347.**[5]**L. I. Strukov and A. F. Timan,*Mathematical expectation of continuous functions of random variables, smoothness, and variance*, Sibirsk. Mat. Ž.**18**(1977), no. 3, 658–664, 719 (Russian). MR**0454471****[6]**A. F. Timan,*Theory of approximation of functions of a real variable*, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR**0192238**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766520-7

Keywords:
Bernstein polynomial,
moduli of smoothness

Article copyright:
© Copyright 1985
American Mathematical Society