*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be the error of the best uniform approximation to a Real function on the
Interval by Real Polynomials of degree at most . If

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

**References**

Bulanov, A. P. ``Asymptotics for the Best Rational Approximation of the Function Sign .''
*Mat. Sbornik* **96**, 171-178, 1975.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/brnstn/brnstn.html

Gonchar, A. A. ``Estimates for the Growth of Rational Functions and their Applications.'' *Mat. Sbornik*
**72**, 489-503, 1967.

Newman, D. J. ``Rational Approximation to .'' *Michigan Math. J.* **11**, 11-14, 1964.

Petrushev, P. P. and Popov, V. A. *Rational Approximation of Real Functions.* New York: Cambridge University Press, 1987.

Stahl, H. ``Best Uniform Rational Approximation of on .'' *Russian Acad. Sci. Sb. Math.* **76**, 461-487, 1993.

Varga, R. S. *Scientific Computations on Mathematical Problems and Conjectures.* Philadelphia, PA: SIAM, 1990.

Varga, R. S. and Carpenter, A. J. ``On a Conjecture of S. Bernstein in Approximation Theory.'' *Math. USSR
Sbornik* **57**, 547-560, 1987.

Varga, R. S.; Ruttan, A.; and Carpenter, A. J. ``Numerical Results on Best Uniform Rational Approximations to
on . *Math. USSR Sbornik* **74**, 271-290, 1993.

Vjacheslavo, N. S. ``On the Uniform Approximation of by Rational Functions.'' *Dokl. Akad. Nauk SSSR*
**220**, 512-515, 1975.

© 1996-9

1999-05-26