# Algebraic polynomial of best approximation

A polynomial deviating least from a given function. More precisely, let a measurable function $ f(x) $
be in $ L _ {p} [a, b] $(
$ p \geq 1 $)
and let $ H _ {n} $
be the set of algebraic polynomials of degree not exceeding $ n $.
The quantity

$$ \tag{* } E _ {n} (f) _ {p} = \inf _ {P _ {n} (x) \in H _ {n} } \| f (x) - P _ {n} (x) \| _ {L _ {p} [ a , b ] } $$

is called the best approximation, while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in $ L _ {p} [a, b] $. Polynomials which deviate least from a given continuous function in the uniform metric ( $ p = \infty $) were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 [1]. The existence of algebraic polynomials of best approximation was established by E. Borel [2]. Chebyshev proved that $ P _ {n} ^ {0} (x) $ is an algebraic polynomial of best approximation in the uniform metric if and only if Chebyshev alternation occurs in the difference $ f(x) - P _ {n} ^ {0} (x) $; in this case such a polynomial is unique. If $ p > 1 $, the algebraic polynomial of best approximation is unique due to the strict convexity of the space $ L _ {p} $. If $ p = 1 $, it is not unique, but it has been shown by D. Jackson [3] to be unique for continuous functions. The rate of convergence of $ E _ {n} {(f) } _ {p} $ to zero is given by Jackson's theorems (cf. Jackson theorem).

In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say $ m $. If the number of variables $ m \geq 2 $, an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.

#### References

[1] | P.L. Chebyshev, "Questions on smallest quantities connected with the approximate representation of functions (1859)" , Collected works , 2 , Moscow-Leningrad (1947) pp. 478; 152–236 (In Russian) |

[2] | E. Borel, "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars (1905) |

[3] | D. Jackson, "A general class of problems in approximation" Amer. J. Math. , 46 (1924) pp. 215–234 |

[4] | A.L. Garkavi, "The theory of approximation in normed linear spaces" Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 (In Russian) |

#### Comments

Instead of the long phrase "algebraic polynomial of best approximation" one also uses the shorter phrase "best algebraic approximationbest algebraic approximation" , which is not to be confused with the phrase "best approximation" for the least error $ E _ {n} (f) _ {p} $.

#### References

[a1] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |

[a2] | E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff |

[a3] | G.W. Meinardus, "Approximation von Funktionen und ihre numerische Behandlung" , Springer (1964) pp. Chapt. 1, §5 |

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Algebraic polynomial of best approximation.

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