Chebyshev constant

A numerical invariant $ \tau = \tau ( E) $ 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)

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)