Potential Theoretic Methods Research Articles

Rational Approximation with Varying Weights, II

We consider two problems concerning uniform approximation by weighted rational functions {wnrn}∞n=1, wherern=pn/qnhas real coefficients, degpn⩽[αn] and degqn⩽[βn], for givenα>0 andβ⩾0. Forw(x):=exwe show that on any interval [0, a] witha∈(0, â(α, β)), every real-valued functionf∈C([0, a]) is the uniform limit of some sequence {wnrn}. An implicit formula forâ(α, β) was given in the first part of this series of papers; in particular,â(1, 1)=2π. Forw(x):=xθwithθ>1 we show that uniform approximation of real-valuedf∈C([b, 1]) on [b, 1] by weighted rationalswnrnis possible for anyb∈(b(θ; α, β),1), whereb(θ; α, β) was also found in Part I; in particular,b(θ; 1, 1)=tan4((π/4)((θ−1)/θ)). Both of the mentioned results are sharp in the sense that approximation is no longer possible ifâis replaced byâ+εorbis replaced byb−εwithε>0. We use potential theoretic methods to prove our theorems.

Density of zeros of some orthogonal polynomials

Abstract. In this paper, we study the asymptotic eigenvalue density of largen × n random Hermitian matrices. The eigenvalue density can be interpretedin the context of orthogonal polynomials as the density of zeros. We adopt twoapproaches; the first, using a recent theorem, gives the density of zeros as anintegral representation with the (appropriately scaled) recurrence coefficients asinput. The second makes use of the Coulomb fluid approach pioneered by Dysonwhere the weight with respect to which the polynomials are orthogonal is theinput.The zero density of the Stieltjes-Wigert, q −1 -Hermite, q−Laguerre polynomialsand a constructed set of orthogonal polynomials are obtained. In the last twocases, the density can be expressed in terms of complete and incomplete ellipticintegrals of various kinds.We also compute, in some cases, the effective potentials from the densities. 1. IntroductionIn the application of the theory of n×nrandom matrices, a quantity of central interestis the asymptotic eigenvalue density for large n. If the matrix is complex Hermitian, itcan be shown that the eigenvalue density is the density of zeros of certain polynomialsorthogonal with respect to a positive weight supported on R or a subset of R wherenis the degree.A more conventional technique for computing the asymptotic density is based onpotential theory, and in the physical context, the Coulomb fluid method pioneered byDyson in the 1960s [7]. For an excellent and up-to-date book on potential theoreticmethods, see [11]. For a physical approach to the density, see [4]. This latter techniquerelies on the weight function, w(x), and more precisely, the external potential associ-ated with it, v(x) := −lnw(x). However, if the external potential increases sufficientlyslowly near infinity, such as those of the q

Zeros and Critical Points of Sobolev Orthogonal Polynomials

Using potential theoretic methods we study the asymptotic distribution of zeros and critical points of Sobolev orthogonal polynomials, i.e., polynomials orthogonal with respect to an inner product involving derivatives. Under general assumptions it is shown that the critical points have a canonical asymptotic limit distribution supported on the real line. In certain cases the zeros themselves have the same asymptotic limit distribution, while in other cases we can only ascertain that the support of a limit distribution lies within a specified set in the complex plane. One of our tools, which is of independent interest, is a new result on zero distributions of asymptotically extremal polynomials. Our results are illustrated by numerical computations for the case of two disjoint intervals. We also describe the numerical methods that were used.

Commentary on "From local to global in quasiconformal structures".

Commentary on "From local to global in quasiconformal structures".

Asymptotic distribution of the zeros of weighted fibonacci polynomials

Exploiting the relation between the classical Fibonacci polynomials {F n(z)} and a certain weighted Faber polynomials {B n(z,g)} associated with a domain E in complex plane and a weight function g(z), we define weighted Fibonacci polynomials in complex domain. Applying fundamental properties of weighted Faber polynomials, we extend basic properties of Fibonacci polynomials to complex plane. Using potential theoretic methods, we determine the asymptotic distribution of the zeros of the weighted Fibonacci polynomials.

Schauder and LPEstimates for Parabolic systems via campanato spaces

The following note deals with classical Schauder and L{sup p} estimates in the setting of parabolic systems. For the heat equation these estimates are usually obtained via potential theoretic methods, i.e., by studying the fundamental solution, and for the elliptic case. For systems, however, it has become customary to base both Schauder and L{sup p} theory on Campanato`s technique. For the elliptic case this is explained in and (see also the references therein to Campanato`s original papers) and for the stationary Stokes system in. The purpose of this note is to show how this technique can be applied to parabolic systems. In many ways it is very similar to Giaquinta`s exposition-in spirit as well as in detail. However, the parabolic case dose offer certain peculiarities, most notably the global Schauder estimates, where we have to deal with a compatibility condition on the data. 13 refs.

Asymptotic distribution of the zeros of Faber polynomials

AbstractUsing potential theoretic methods and exploiting the connection with eigenvalues of Toeplitz matrices, we investigate the limiting behaviour of zeros of Faber polynomials generated by a Laurent series. Our results build upon fundamental work of J. L. Ullman. For example, we show that if E is a compact set with simply connected complement and connected interior whose boundary is either (i) not a piecewise analytic curve or (ii) a piecewise analytic curve but with a singularity other than an outward cusp, then the equilibrium distribution for E is a limit measure of the sequence of normalized zero counting measures for the Faber polynomials associated with E.

On the riesz charge of the lower envelope of δ-subharmonic functions

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more δ-subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.

The sedimentation of a small particle through a fluid-filled pore of finite length

This paper investigates the axisymmetric sedimentation of a small slowly rotating and translating particle through a fluid-filled circular pore of finite length which communicates with two half-space chambers of fluid. In the quasi-steady Stokes approximation the particle is modelled by either a rotlet or Stokeslet, and potential-theoretic methods are used to reduce each problem to the solution of coupled infinite systems of linear equations. The numerical solutions of these sets of equations are used to compute approximations to the resistive torque and drag experienced by the particle.

Plane interior boundary-value problems in microcontinuum fluid mechanics

This paper examines the fundamental question of existence and uniqueness of solutions of two plane interior boundary-value problems encountered in the field of microcontinuum fluid mechanics. These problems are analyzed using potential theoretic methods and with the aid of singular integral equations. The results obtained have their counterparts in the classical Navier-Stokes theory.

On the convergence of certain singular integrals

Hardy (3) provides one of the earliest rigorous discussions of the convergence of Riemann multiple integrals of real functions with a general type of algebraic singularity. More recently, and with the advent of Lebesgue integration, various authors such as Garsia(2) have investigated singular integrals involving more general measurable functions. In this paper we shall develop an indirect potential theoretic method to obtain convergence criteria for a class of multiple integral that seems to have been neglected to date.

On the closed ideals inA(W)

scription of the closed ideals in the disc algebra is shown to apply to an ideal whose hull meets the boundary of the domain in a finite union of analytic arcs. The canonical factorization into inner and outer functions in the disc is replaced by a potential theoretic decomposition theorem, thus allowing essentially the same description to be carried over. The basically local nature of the problem is used to reduce it to the previously known ideal theory of a compact bordered Riemann surface. This reduction is facilitated by a factorization theorem that is proved by potential theoretic methods.

The linear travelling-wave induction problem

The problem considered is that of evaluating the fields induced in a long conducting cylinder, with arbitrary cross-sectional shape and arbitrary constant scalar conductivity and permeability, placed in the field of a constant current source. The spatial-time variations of the source is described by a single wave or a spectrum of waves travelling parallel to the axis. The relative velocity between the source and the cylinder is in the same direction. It is shown that for a quasi-static magnetic field, the induced fields (at the cylinder surface) are conveniently decomposed into TM and TE components that are unidirectionaly coupled. Potential theoretic methods are used to derive 1-dimensional Fredholm integral equations of the second kind for the fields of each mode. An example is given to demonstrate the details of numerical solution. The dimensionless parameters affecting the field distribution are indicated.

A self-consistent theory of steady-state lamellar solidification in binary eutectic systems

The potential theoretic methods developed recently at NRL for solving the diffusion equation are applied to the free-boundary problem which describes lamellar solidification in binary eutectic systems. By using these techniques, the original free-boundary problem is reduced to a set of coupled nonlinear integro-differential equations which when solved yield the shape of the solid/liquid interface and the solute concentration on the interface. A simplified version of the general integro-differential equations is derived by assuming that (1) the solute diffusion length is large compared with the lamellar spacing, and (2) the solid/liquid interface is approximately isothermal. Numerical solutions to the approximate equations are obtained and are utilized to assess the effect of interface curvature on the interfacial solute concentrations, and to check the new theory for consistency with experiment.

Boundary value problems in generalized biaxially symmetric potential theory

Interior boundary value problems are solved for the operator of generalized biaxially symmetric potential theory. The boundary conditions consist of Dirichlet data on the nonsingular part of the boundary and Dirichlet data or growth restrictions on the singular hyperplanes, depending on the values of parameters of the operator. Continuation of solutions beyond the singular hyperplanes is considered, yielding an improvement of a result of Huber. Potential theoretic methods are used for the investigation.

The value distribution of harmonic mappings between Riemanniann-spaces

We are concerned with the value distribution of a mapping of an open Riemanniann-space (n≧ 3) into a Riemanniann-space. The value distribution theory of an analytic mapping of Riemann surfaces was initiated by S. S. Chern [1] and developed mainly by L. Sario [8], [9], [10], [11], and then by H. Wu [14], [15]. The most crucial part in Sario’s theory is the introduction of a kernel function on an arbitrary Riemann surface to describe appropriately the proximity of two points. His method indicates that the potential theoretic method is one of the powerful methods in the value distribution theory.

Sario’s Potentials and Analytic Mappings

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

The charge density near a sharp point on a conductor

ABSTRACTA brief discussion is given of attempts that have been made to justify, in terms of electrostatic principles, the conjecture that surface charge on a conductor tends to infinity towards a convex sharp point and to zero towards a concave point, and it is concluded that the problem has not hitherto been solved. A new attempt is then made using potential-theoretic methods, and the problem is solved with a fair degree of generality. The limitations are of a kind familiar in this type of analysis and, roughly speaking, concern the degree of convexity of the conducting surface.

The Kernel Function and Conformal Mapping

Orthogonal functions The kernel function and associated minimum problems The invariant metric and the method of the minimum integral Kernel functions and Hilbert space Representation of the classical domain functions Canonical conformal transformations Orthogonalization over the boundary Variational methods Existence proofs Partial differential equations Functions of two complex variables and pseudoconformal mapping Generalization of potential-theoretical methods and certain subclasses of functions Bibliography Index.