Extremal properties of polynomials

Properties of algebraic, trigonometric or generalized polynomials that distinguish them as solutions of some extremal problems.

For example, the Chebyshev polynomials $ T _ {n} ( x) = \cos ( n { \mathop{\rm arc} \cos } x ) = 2 ^ {n-} 1 x ^ {n} + \dots $ have minimal norm in the space $ C ( [ - 1 , 1 ] ) $ among all algebraic polynomials of degree $ n $ with leading coefficient $ 2 ^ {n-} 1 $( P.L. Chebychev, 1853); therefore they are the solution of the extremal problem

$$ \max _ {x \in [ - 1 , 1 ] } | 2 ^ {n-} 1 x ^ {n} + a _ {1} x ^ {n-} 1 + \dots + a _ {n} | \rightarrow \inf _ {a = ( a _ {1} \dots a _ {n} ) } . $$

In other words, the polynomial $ T _ {n} $ differs least from zero in the space $ C ( [ - 1 , 1 ] ) $ among all polynomials of degree $ n $ with leading coefficient equal to $ 2 ^ {n-} 1 $.

Extremal problems in spaces of polynomials are mainly studied in the spaces $ L _ {p} ( [ a , b ] ) $, $ 1 \leq p \leq \infty $. In this context most of the available results are connected with the cases $ p = 1 $, 2 and $ \infty $( the metric of $ C $). In particular, these metrics are used to find the explicit form of the polynomials that differ least from zero. In the metric of $ L _ {1} $ these are the Chebychev polynomials of the second kind, in the metric of $ L _ {2} $ one obtains the Legendre polynomials; concerning the metric of $ C $ see above. The set of classical orthogonal polynomials deviating least from zero in a weighted $ L _ {2} $- space (Laguerre polynomials; Hermite polynomials; Jacobi polynomials; etc.) has also been described.

E.I. Zolotarev (1877) considered the question of determining polynomials of the form $ x ^ {n} + \sigma x ^ {n-} 1 + a _ {2} x ^ {n-} 2 + \dots + a _ {n} $( with two fixed leading coefficients) that deviate least from zero in the metric of $ C $. He found a one-parameter family of polynomials that solved this problem, and expressed them in terms of elliptic functions.

The Chebychev polynomials are extremal in the problem of an inequality for the derivatives; namely, the exact Markov inequality (where $ P _ {n} $ is a polynomial of degree $ \leq n $)

$$ \tag{* } \| P _ {n} ^ {(} k) \| _ {C ( [ - 1 , 1 ] ) } \leq | T _ {n} ^ {(} k) ( 1) | \cdot \| P _ {n} \| _ {C ( [ - 1 , 1 ] ) } $$

holds, with equality for $ T _ {n} $. Inequality (*) was proved by A.A. Markov (1889) for $ k = 1 $, and by V.A. Markov (1892) for all other values of $ k $. Concerning a similar inequality for trigonometric polynomials see Bernstein inequality.

Some extremal properties of algebraic and trigonometric polynomials in the uniform metric carry over to Chebychev systems of functions (see [2]). Concerning the theory of extremal problems and extremal properties of polynomials see [6].

References

[1]	P.L. Chebychev, "Oeuvres" , 1–2 , Chelsea, reprint (No date)
[2]	S.N. Bernshtein, "Leçons sur les propriétés extrémales de la meilleure approximation des fonctions analytiques d'une variable réelle" , Chelsea, reprint (1970) (Translated from Russian)
[3]	V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[4]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[5]	E.V. Voronovskaya, "The functional method and its applications" , Amer. Math. Soc. (1970) (Translated from Russian)
[6]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)

References

[a1]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[a3]	T.J. Rivlin, "The Chebyshev polynomials" , Wiley (1974)
[a4]	H.S. Shapiro, "Topics in approximation theory" , Springer (1971)
[a5]	G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)