Integral estimates of lengths of level lines of rational functions and zolotarev’s problem

Integral estimates of lengths of level lines (lemniscates) of rational functions of a complex variable are obtained. These estimates are related to the problem of separation of compact sets by rational functions and to Zolotarev’s problem.     

On the Harmonic Measure and Capacity of Rational Lemniscates

We study the lemniscates of rational maps. We prove a reflection principle for the harmonic measure of rational lemniscates and we give estimates for their capacity and the capacity of their components. Also, we prove a version of Schwarz’s lemma for the capacity of the lemniscates of proper holomorphic functions.     

Volume of Polynomial Lemniscates in C n

Kiselman's method permitting one to estimate the volume of sublevel sets of a plurisubharmonic function f in C ⁿ with the aid of the Lelong number of f is applied to polynomial lemniscates in C ⁿ . In particular, it yields estimates from below of the volume of polynomial lemniscates which completes recent results of Cuyt, Driver and Lubinsky.     

On the size of multivariate polynomial lemniscates and the convergence of rational approximants

In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest.     

Lemniscates 3D: a CAGD primitive?

A 3D lemniscate is an implicitly given surface which generalizes the well-known Bernoulli lemniscates curves and the Cassini ovals in 2D. It is characterized by placing a finite number of points in space (the foci) and choosing a constant (radius), its algebraic degree is twice the number of foci and it is always contained in the union of certain spheres centered at the foci. The distribution of the foci gives a rough idea of the 3D shapes that could be modeled with any of the connected components of the lemniscate. The position of the foci can be used to stretch and to produce knoblike features. Given a set of foci, for a small radius the lemniscate consists of a number of spherelike surfaces centered at the foci which do not touch each other. As the radius increases the disconnected pieces coalesce producing interesting surfaces. In order to make 3D lemniscates a potentially useful primitive for CAGD it is necessary to control the coalescing/splitting of the connected components of the lemniscate while we move the foci and change the radius, simultaneously. In this paper we offer tools towards this control. We look closely at the case of four noncoplanar foci. AMS subject classification 65D05, 65D17, 65D18This work was partially supported by grant G97 000651 of Fonacit, Venezuela.     

An extremal problem for the derivative of a rational function

Erd?s’ well-known problem on the maximum absolute value of the derivative of a polynomial on a connected lemniscate is extended to the case of a rational function. Moreover, under the assumption that certain lemniscates are connected, a sharp upper bound for the absolute value of the derivative of a rational function at any point in the plane different from the poles is found. The role of the extremal function is played by an appropriate Zolotarev fraction.     

Chebyshev polynomials on a system of curves

This paper is devoted to the problem of how close can one get with the n-th Chebyshev numbers of a compact set ?? to the theoretical lower bound cap(??) ⁿ . It is shown that for a system of m ?? 2 analytic curves, there is always a subsequence for which the Chebyshev numbers are at least (1 + ??)cap(??) ⁿ , while for another subsequence they are at most (1 + O(n ^?1/(m?1)))cap(??) ⁿ . It is also shown that the last estimate is optimal. We also discuss how well a system of curves can be approximated by lemniscates in Hausdorff metric. The proofs are based on potential theoretical arguments. Simultaneous Diophantine approximation of harmonic measures lies in the background. To achieve the correct rate, a perturbation of the multi-valued complex Green??s function is introduced which makes the n-th power of its exponential single-valued and which allows the construction of Faber-like polynomials on multiply connected domains.     

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 Erdélyi. 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. Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.     

On the Approximation of a Continuum by Lemniscates

For an arbitrary continuum E in the complex plane with connected complement Ω we study the rate of approximation of ∂E from outside by lemniscates in terms of level lines of a conformal mapping of Ω onto the exterior of the unit disk.     

The logarithmic energy of zeros and poles of a rational function

On assuming that certain lemniscates of a rational function are connected, we establish some sharp inequality that involves the logarithmic energy of a discrete charge concentrated at the zeros and poles of this function and the absolute values of its derivatives at these points. The equality in this estimate is attained for specially arranged zeros and poles of a suitable Zolotarev fraction and for special distributions of charge.     

On the degree of convergence of lemniscates in finite connected domains

For an arbitrary bounded closed set E in the complex plane with complement Ω of finite connectivity, we study the degree of convergence of the lemniscates in Ω.     

A property of the planar measure of the lemniscates

In this paper, we establish the following conjecture: There exists a constant K such that every lemniscate E(α,c), α∈Cn, c>0, contains a disk B(α,c) with μ(E(α,c))?Kμ(B(α,c)), where μ is the planar measure. We prove this conjecture for any family of lemniscates with at the most three foci and for any family of lemniscates where its foci satisfy a suitable condition.     

Convergence of Row Sequences of Simultaneous Padé–Faber Approximants

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.     

Application du prolongement analytique au problème inverse du potentiel

Sommaire Le but de cet article est établir quelques résultats nouveaux sur le problème inverse du potentiel newtonien. Nous démontrons deux théorèmes d'unicité: pour les polyédres convexes dansR ⁿ et pour les lemniscates dansR ². L'instrument principal est un lemme basé sur une idée de V. Kondrachkov rarement utilisé malgré sa puissance. Nous montrons son efficacité en liaison avec la méthode du prolongement analytique des potentiels.

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The goal of this paper is to establish some new results in the inverse Newtonian potential problem. We prove two uniqueness theorems: for convex polyhedra inR ⁿ and for lemniscates inR ². The main tool is a lemma based upon an idea of V. Kondrashkov which, though powerful, is rarely used. We show its efficiency applied together with the method of analytic continuation of potentials.

     

On the Rate of Approximation of Closed Jordan Curves by Lemniscates

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve Γ in the complex plane {z} by lemniscates generated by polynomials P(z). In the present paper, we obtain quantitative upper bounds for the least deviations H _n (Γ) (in this metric) from the curve Γ of the lemniscates generated by polynomials of a given degree n in terms of the moduli of continuity of the conformal mapping of the exterior of Γ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of Γ. For the case in which the curve Γ is analytic, we prove that H _n (Γ) = O(q _n ), 0 ≤ q = q(Γ) < 1, n → ∞.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 861–876.Original Russian Text Copyright ©2005 by O. N. Kosukhin.     

The Sequence-covering Compact Images of Paracompact Locally Compact Spaces

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Instability for harmonic foliations on compact homogeneous spaces

In this paper we discuss the instability of harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces. We obtain a sufficient condition for a harmonic foliation to be unstable on compact submanifolds in a Euclidean space and on compact isotropy irreducible homogeneous spaces. We also classify compact symmetric spaces which have no non-trivial stable harmonic foliation.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

About positive Dunford-Pettis operators on Banach lattices

We give several characterizations of Banach lattices on which each positive Dunford-Pettis operator is compact. As consequences, we obtain new sufficient and necessary conditions under which a norm of a Banach lattice is order continuous, a positive weakly compact operator is compact and the dual operator of a positive Dunford-Pettis operator is Dunford-Pettis.     

On products with the unit interval

We investigate the question which compact convex sets are homeomorphic to their product with the unit interval. We prove it in particular for the space of probability measures on any infinite scattered compact space and for the half-ball of a non-separable Hilbert space equipped with the weak topology. We also show examples of compact spaces for which it is not the case.