E. B. Saff

Logarithmic potentials with external fields

This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; the numerical conformal mapping of simply and doubly connected domains; generalization of the Weierstrass approximation theorem to varying weights; and the determination of convergence rates for best approximating rational functions.

Where Does the Sup Norm of a Weighted Polynomial Live? (A Generalization of Incomplete Polynomials)

A characterization is given of the sets supporting the uniform norms of weighted polynomials [w(x)]nPn(x), wherePn is any polynomial of degree at mostn. The (closed) support ∑ ofw(x) may be bounded or unbounded; of special interest is the case whenw(x) has a nonempty zero setZ. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of ∑ —Z. One main result of this paper states that there is a unique compact set (independent ofn andPn) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights [w(x)]n is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.

Asymptotics for minimal discrete energy on the sphere

We investigate the energy of arrangements of N points on the surface of the unit sphere Sd in Rd+1 that interact through a power law potential V = 1/rs, where s > 0 and r is Euclidean distance. With Ed(s,N) denoting the minimal energy for such N-point arrangements we obtain bounds (valid for all N) for Ed(s, N) in the cases when 0 < s < d and 2 ≤ d < s. For s = d, we determine the precise asymptotic behavior of Ed(d,N) as N → ∞. As a corollary, lower bounds are given for the separation of any pair of points in an N-point minimal energy configuration, when s ≥ d ≥ 2. For the unit sphere in R3 (d = 2), we present two conjectures concerning the asymptotic expansion of E2(s,N) that relate to the zeta function ζL(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of ζL(s) when 0 < s < 2 (the divergent case).

Constrained Energy Problems with Applications to Orthogonal Polynomials of a Discrete Variable

Given a positive measure Σ with gs > 1, we write Με ℳΣ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λΣw that minimizes the weighted logarithmic energy over the class ℳΣ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λΣw As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials.

On the zeros and poles of Padé approximants toez

In this paper, we study the location of the zeros and poles of general Pade approximats toe z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.

On the distribution of zeros of polynomials orthogonal on the unit circle

Abstract Let { Φ n } be a system of monic polynomials orthogonal on the unit circle with respect to a positive Borel measure μ. It is shown that under fairly mild conditions on the reflection coefficients, Φ n (0), the zeros of a subsequence of { Φ n } are asymptotically uniformly distributed on some circle. The radius of this circle can be found using a Cauchy-Hadamard-type formula. We also characterize the measures μ with the property that the L dμ 2 best polynomial approximants to every function f ϵ H 2 ( μ ) converge uniformly on every compact subset of the open unit disc at a geometrically fast rate.

On the Eneström-Kakeya theorem and its sharpness

Abstract A new proof, based on the Perron-Frobenius theory of nonnegative matrices, is given of a result of Hurwitz on the sharpness of the classical Enestrom-Kakeya theorem for estimating the moduli of the zeros of a polynomial with positive real coefficients. It is then shown (Theorem 2) that the zeros of a particular set of polynomials fill out the Enestrom-Kakeya annulus in a precise manner, and this is illustrated by numerical results in Fig. 1.

Where does the ^{}-norm of a weighted polynomial live?

A characterization is given of the sets supporting the uniform norms of weighted polynomials [w(x)]nPn(x), wherePn is any polynomial of degree at mostn. The (closed) support ∑ ofw(x) may be bounded or unbounded; of special interest is the case whenw(x) has a nonempty zero setZ. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of ∑ —Z. One main result of this paper states that there is a unique compact set (independent ofn andPn) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights [w(x)]n is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.

Zero-Free Parabolic Regions for Sequences of Polynomials

In this paper, we show that certain sequences of polynomials

Zeros of Sections of Power Series

\{ p_k (z)\} _{k = 0}^n