Interpolatory properties of best L2-approximants

AbstractLet ƒbe a continuous function and sn be the polynomial of degree at most n of best L2(μ)-approximation to ƒon [-1,1]. Let Zn(ƒ):=\s{xϵ[-1,1]:ƒ(x)−sn(x) = 0\s}. Under mild conditions on the measure μ, we prove that ∪ Zn(ƒ) is dense in [-1,1]. This answers a question posed independently by A. Kroó and V. Tikhomiroff. It also provides an analogue of the results of Kadec and Tashev (for L∞) and Kroó and Peherstorfer (for L1) for least squares approximation

Local behaviour of the error in the Bergman kernel method for numerical conformal mapping

AbstractLet Ω be a simply-connected domain in the complex plane, let ζ ϵ Ω and let K(z, ζ) denote the Bergman kernel function of Ω with respect to ζ. Also, let Kn(z, ζ) denote the nth-degree polynomial approximation to K(z, ζ), given by the classical Bergman kernel method, and let πn denote the corresponding nth-degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. Finally, let B be any subdomain of Ω. In this paper we investigate the two local errors ‖K(·, ζ)−Kn(·, ζ)|L2(B), |f′ζ − π′n|L2(B), and compare their rates of convergence with those of the corresponding global errors with respect to L2(Ω). Our results show that if ∂B contains a subarc of ∂Ω, then the rates of convergence of the local errors are not substantially different from those of the global errors

Estimating the Argument of Some Analytic Functions

AbstractWe consider a class of analytic functions that are closely related to approximate conformal mappings of simply connected domains onto the unit disk. Using a result of Warschawski, we improve upon our estimates of the argument in the considered class for the important case when the mapping is nearly circular. This new estimate is asymptotically sharp

Rational Approximation with Locally Geometric Rates

AbstractWe investigate the rate of pointwise rational approximation of functions from two classes. The distinguishing feature of these classes is the essentially faster convergence of the best uniform rational approximants versus best uniform polynomial approximants. It is known that for piecewise analytic functions “near best” polynomials converging geometrically fast at every point of analyticity of the function exist. Here we construct rational approximants enjoying similar properties. We also show that our construction yields rates of convergence that are, in a certain sense, best possible

Menke points on the real line and their connection to classical orthogonal polynomials

AbstractWe investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1]

Mesh ratios for best-packing and limits of minimal energy configurations

For N-point best-packing configurations ωN on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ρN,A), which is the quotient of the covering radius of ωN relative to A and the minimum pairwise distance between points in ωN . For best-packing configurations ωN that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ωN,A)≤1 and this bound can be attained even for the sphere. In the particular case when N=5 on S1 with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid ω∗5, that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, γ(ω∗5,S2)=1

Behavior of Lagrange interpolants to the absolute value function in equally spaced points

We find the weak star limit of the sequence of normalized counting measures of the zeros of the Lagrange interpolants to fs(x) = |x − s|(−1 <s< 1) associated with equidistant nodes on [−1, 1]. We use this to establish the exact region in which the Lagrange interpolants converges geometrically

Minimal Riesz energy on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x-y|^(-s) with d-2 <= s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S^d is determined. The special case s = d-2 yields interesting phenomena, which we investigate in detail. A weak* asymptotic analysis is provided as s goes to (d-2)^+.Comment: 42 pages, 2 figure

First Colonization of a Spectral Outpost in Random Matrix Theory

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitean matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N^(-2 h) where 1/h = 2 nu+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general all these newborn zeroes approach the point of nonregularity at the rate N^(-h) whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term: in particular the transition between K and K+1 eigenvalues is analyzed in detail.Comment: 32 pages, 8 figures (typo corrected in Formula 4.13); some reference added and minor correction

Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit

We consider the double scaling limit in the random matrix ensemble with an external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.Comment: 36 pages, 9 figure