Jackson's theorem is a statement about the error of the best uniform approximation to a Real Function on by Real Polynomials of degree at most . Let
be of bounded variation in and let and denote the least upper bound of and the total variation of
in , respectively. Given the function

(1) |

(2) |

(3) |

Bernstein strengthened Jackson's theorem to

(4) |

(5) |

(6) |

**References**

Cheney, E. W. *Introduction to Approximation Theory.* New York: McGraw-Hill, 1966.

Jackson, D. *The Theory of Approximation.* New York: Amer. Math. Soc., p. 76, 1930.

Rivlin, T. J. *An Introduction to the Approximation of Functions.* New York: Dover, 1981.

Sansone, G. *Orthogonal Functions, rev. English ed.* New York: Dover, pp. 205-208, 1991.

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1999-05-25