# Difference between revisions of "Chebyshev approximation"

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''uniform approximation'' | ''uniform approximation'' | ||

− | Approximation of a continuous function | + | Approximation of a continuous function $f$ defined on a set $M$ by functions $S$ from a given class of functions, where the measure of approximation is the deviation in the uniform metric |

− | + | $$\rho(f,S)=\sup_{x\in M}|f(x)-S(x)|.$$ | |

− | P.L. Chebyshev in 1853 [[#References|[1]]] raised and studied the problem of best uniform approximation of a continuous function by algebraic polynomials of degree not exceeding | + | P.L. Chebyshev in 1853 [[#References|[1]]] raised and studied the problem of best uniform approximation of a continuous function by algebraic polynomials of degree not exceeding $n$. For this problem, and also for the more general problem concerning best uniform approximation of a function by rational functions, he obtained fundamental results, and at the same time laid the foundations for the theory of [[Best approximation|best approximation]]. |

====References==== | ====References==== |

## Latest revision as of 08:42, 19 April 2014

*uniform approximation*

Approximation of a continuous function $f$ defined on a set $M$ by functions $S$ from a given class of functions, where the measure of approximation is the deviation in the uniform metric

$$\rho(f,S)=\sup_{x\in M}|f(x)-S(x)|.$$

P.L. Chebyshev in 1853 [1] raised and studied the problem of best uniform approximation of a continuous function by algebraic polynomials of degree not exceeding $n$. For this problem, and also for the more general problem concerning best uniform approximation of a function by rational functions, he obtained fundamental results, and at the same time laid the foundations for the theory of best approximation.

#### References

[1] | P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 23–51 (In Russian) |

[2] | R.S. Guter, L.D. Kudryavtsev, B.M. Levitan, "Elements of the theory of functions" , Moscow (1963) (In Russian) |

#### Comments

See also [a1], especially Chapt. 3, and [a2], Section 7.6. For an obvious reason, Chebyshev approximation is also called best uniform approximation.

#### References

[a1] | E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) |

[a2] | P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126 |

**How to Cite This Entry:**

Chebyshev approximation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_approximation&oldid=12729