Chebyshev constant
A numerical invariant
of a compact set E
in the complex plane that is used in the theory of best approximation.
Let K _ {n} be the class of all polynomials
p _ {n} ( z) = z ^ {n} + c _ {1} z ^ {n-1} + \dots + c _ {n}
of degree n , and let
M ( p _ {n} ) = \max \{ {| p _ {n} ( z) | } : {z \in E } \} ,
m _ {n} = \inf \{ {M ( p _ {n} ) } : {p _ {n} \in K _ {n} } \} ,\ \tau _ {n} = {m _ {n} } ^ {1/n} .
There exists a polynomial t _ {n} ( z) \in K _ {n} for which M ( t _ {n} ) = m _ {n} ; it is called the Chebyshev polynomial for E . Moreover, the limit
\lim\limits _ {n \rightarrow \infty } \tau _ {n} = \tau
exists, and is called the Chebyshev constant for E .
Restricting oneself to the class \widetilde{K} _ {n} of all polynomials
\widetilde{p} _ {n} ( z) = z ^ {n} + \dots + \widetilde{c} _ {n}
all zeros of which lie in E , one obtains corresponding values \widetilde{m} _ {n} , \widetilde \tau _ {n} , \widetilde \tau and a polynomial \widetilde{t} _ {n} ( z) for which M ( \widetilde{t} _ {n} ) = \widetilde{m} _ {n} (it is also called the Chebyshev polynomial).
It is known that \tau = \widetilde \tau = C ( E) = d , where C ( E) is the capacity of the compact set E , and d is its transfinite diameter (cf., for example, [1]).
The concept of the Chebyshev constant generalizes to compact sets E in higher-dimensional Euclidean spaces \mathbf R ^ {m} starting from potential theory. For a point x \in \mathbf R ^ {m} , let
H ( | x | ) = \left \{ \begin{array}{ll} \mathop{\rm ln} \frac{1}{| x | } & \textrm{ for } m = 2 , \\ \frac{1}{| x | ^ {m-2} } & \textrm{ for } m \geq 3 , \\ \end{array} \right .
be the fundamental solution of the Laplace equation, and for a set ( x _ {j} ) _ {j=1} ^ {n} \subset E , let
\sigma _ {n} ( E) = \sup \left \{ {\min \left \{ { \frac{1}{n} \sum _ { j= 1} ^ { n } H ( | x - x _ {j} | ) } : {x \in E } \right \} } : {( x _ {j} ) _ {j=1} ^ {n} \subset E } \right \} .
Then for m = 2 one obtains the relation
\tau = \widetilde \tau = C ( E) = \mathop{\rm exp} \left ( - \lim\limits _ {n \rightarrow \infty } \ \sigma _ {n} ( E) \right ) ,
and for m \geq 3 one obtains (cf. [2]):
\tau = C ( E) = \frac{1}{\lim\limits _ {n \rightarrow \infty } \sigma _ {n} ( E) } .
References
[1] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
[2] | L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) |
Comments
References
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_constant&oldid=52089