A numerical invariant of a compact set in the complex plane that is used in the theory of best approximation.
Let be the class of all polynomials
of degree , and let
There exists a polynomial for which ; it is called the Chebyshev polynomial for . Moreover, the limit
exists, and is called the Chebyshev constant for .
Restricting oneself to the class of all polynomials
all zeros of which lie in , one obtains corresponding values and a polynomial for which (it is also called the Chebyshev polynomial).
The concept of the Chebyshev constant generalizes to compact sets in higher-dimensional Euclidean spaces starting from potential theory. For a point , let
be the fundamental solution of the Laplace equation, and for a set , let
Then for one obtains the relation
and for one obtains (cf. ):
|||G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)|
|||L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967)|
|[a1]||M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)|
|[a2]||J.L. Walsh, "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc. (1956)|
Chebyshev constant. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Chebyshev_constant&oldid=17435