Chebyshev Constants Research Articles

Polarization constants for products of linear functionals over $\mathbb{R}^2$ and $\mathbb{C}^2$ and Chebyshev constants of the unit sphere

Polarization constants for products of linear functionals over $\mathbb{R}^2$ and $\mathbb{C}^2$ and Chebyshev constants of the unit sphere

Rendezvous numbers in normed spaces

In previous papers, we used abstract potential theory, as developed by Fuglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We considered Chebyshev constants and energies as two variable set functions, and introduced a modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces.Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in all Banach spaces. We also observe the connections of these magical numbers to Chebyshev constants, Chebyshev radius and entropy. Applying the developed notions with the available methods we calculate the rendezvous numbers or rendezvous intervals of certain concrete Banach spaces. In particular, a satisfactory description of the case of Lp spaces is obtained for all p > 0.

Small polynomials with integer coefficients

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdelyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of the integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for the integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for the integer Chebyshev constant. Moreover, all the bounds mentioned can be found numerically by using various extremal point techniques, such as the weighted Leja points algorithm. Applying our results in the classical case of the segment [0, 1], we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials.

Chebyshev Polynomial Iterations and Approximate Solutions of Linear Operator Equations

An iteration scheme for the approximate solution of a linear operator equation in a Banach space is discussed from the viewpoint of Chebyshev polynomials. The optimal rate of convergence is described by numerical characteristics which are similar to (but different from) the classical Chebyshev constants. The abstract results are illustrated by some examples which frequently arise in applications.

Indicators of Growth of Polynomials of Best Uniform Approximation to Holomorphic Functions on Compacta in CN

Let E be a compact and L-regular subset of C N. Siciak has shown that a function ƒ on E has a holomoprhic extension to E R—the interior of the level curve of the Siciak extremal function—if and only if lim sup n → ∞ (sup E |ƒ − p n | 1/ n ) ≤ 1/ R ( R > 1), where p n is a best approximating polynomial to ƒ of degree not greater than n. The aim of this paper is to show that ƒ has a holomorphic extension to E R if for some sequence { p n } of the polynomials of best approximation to ƒ [formula] and if ƒ has such an extension, for all { p n }, there holds [formula]. Here [formula] denotes a norm on the homogeneous terms of degree n in p n and c m ( E), d( E) are some multidimensional counterparts of the logarithmic capacity and the Chebyshev constant, respectively.

Weighted analogues of capacity, transfinite diameter, and Chebyshev constant

For an arbitrary closed subsetE of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant ofE with respect to an admissible weightw onE are introduced. For thew-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an extremal measure whose support is the “smallest” compact set where the sup norm of weighted polynomials “live.” The introduction of a weightw has the advantage that the classical quantities mentioned in the title can be considered for unbounded setsE. Some of the theorems presented are generalizations of the authors' previous results for the case whenE⊂R.

A characterization of weighted Chebyshev constant

A sufficient and necessary condition in order that n th roots of sup norms of weighted polynomials converge to weighted Chebyshev constant will be shown.

Approximation of meromorphic functions by rational functions

Some inequalities proved by Meinardus and Varga, and by Erdös and Reddy, on Chebyshev constants for the function 1 f , f entire and satisfying some conditions, have been improved or extended to functions satisfying a different set of conditions.

Chebyshev Constant for Centered Sets

Using $\lambda$th power means in the case $\lambda \geq 1$ it is proven that the Chebyshev constant of any bounded centered set in a metric space is equal to one-half the topological diameter of the set. Thus the Chebyshev constant for any unit ball of a normed linear space is equal to one for the above means.

Chebyshev constant for centered sets

Using $\lambda$th power means in the case $\lambda \geq 1$ it is proven that the Chebyshev constant of any bounded centered set in a metric space is equal to one-half the topological diameter of the set. Thus the Chebyshev constant for any unit ball of a normed linear space is equal to one for the above means.

Chebyshev constant and Chebyshev points

Using $\lambda$th power means in the case $\lambda \geq 1$, it is proven that the Chebyshev constant for any compact set in ${R_n}$, real Euclidean n-space, is equal to the radius of the spanning sphere. When $\lambda > 1$, the Chebyshev points of order m for all $m \geq 1$ are unique and coincide with the center of the spanning sphere. For the case $\lambda = 1$, it is established that Chebyshev points of order m for a compact set E in ${R_2}$ are unique if and only if the cardinality of the intersection of E with its spanning circle is greater than or equal to three.