Approximation in C^N

This is a survey article on selected topics in approximation theory. The topics either use techniques from the theory of several complex variables or arise in the study of the subject. The survey is aimed at readers having an acquaintance with standard results in classical approximation theory and complex analysis but no apriori knowledge of several complex variables is assumed

Random polynomials and pluripotential-theoretic extremal functions

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related orthonormal basis are independent, identically distributed complex-valued random variables having a very general distribution (which includes both normalized complex and real Gaussian distributions) and we prove results on a.s. convergence of a sequence 1/n log |H_n| pointwise and in L^1_{loc}(C^m) to V_{K,Q}. In addition we obtain results on a.s. convergence of a sequence of normalized zero currents dd^c [1/n log |H_n|] to dd^c V_{K,Q} as well as asymptotics of expectations of these currents. All these results extend to random polynomial mappings and to a more general setting of positive holomorphic line bundles over a compact Kahler manifold

Strong asymptotics for Christoffel functions of planar measures

We prove a version of strong asymptotics of Christoffel functions with varying weights for a general class of sets E and measures in the complex plane. This class includes all regular measures in the sense of Stahl-Totik on regular compact sets E in the plane and even allows varying weights. Our main theorems cover some known results for subsets E of the real line R; in particular, we recover information in the case of E=R with Lebesgue measure dx and weight w(x) = exp(-Q(x)) where Q(x) is a nonnegative, even degree polynomial having positive leading coefficient

Pluripotential Energy

For probability measures $\mu$ on compact subsets of \CC^n we define two functionals $J(\mu)$ and $W(\mu)$ modeled on discrete approximations to $\mu$ and multivariate Vandermonde determinants. We show that these functionals coincide, up to a constant, with the electrostatic energy of $\mu$ defined in a more general setting by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane; i.e., the case $n=1$

A global domination principle for P-pluripotential theory

We prove a global domination principle in the setting of P-pluripotential theory. This has many applications including a general product property for P-extremal functions. The key ingredient is the proof of the existence of a strictly plurisubharmonic P-potential

Zeros of Faber polynomials for Joukowski airfoils

Let $K$ be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for $K$ are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized counting measures of the Faber polynomials for Joukowski airfoils; it is never equal to the potential-theoretic equilibrium measure of $K$. This implies that many of these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of Ullman related to Chebyshev quadrature.Comment: 18 page

The Asymptotics of Optimal Designs for Polynomial Regression

We give the asymptotics for D-optimal (equivalently G-optimal) designs on a compact (possibly complex) design space

Vector Energy and Large Deviation

For d nonpolar compact sets K_1,...,K_d in the complex plane, d admissible weights Q_1,...,Q_d, and a positive semidefinite d x d interaction matrix C with no zero column, we define natural discretizations of the associated weighted vector energy of a d-tuple of positive measures \mu=(\mu_1,...,\mu_d) where \mu_j is supported in K_j and has mass r_j. We have an L^{\infty}-type discretization W(\mu) and an L^2-type discretization J(\mu) defined using a fixed measure \nu=(\nu_1,...,\nu_d). This leads to a large deviation principle for a canonical sequence of probability measures on this space of d-tuples of positive measures if \nu=(\nu_1,...,\nu_d) is a strong Bernstein-Markov measure

Logarithmic potential theory and large deviation

We derive a general large deviation principle for a canonical sequence of probability measures, having its origins in random matrix theory, on unbounded sets $K$ of ${\bf C}$ with weakly admissible external fields $Q$ and very general measures $\nu$ on $K$. For this we use logarithmic potential theory in ${\bf R}^{n}$, $n\geq 2$, and a standard contraction principle in large deviation theory which we apply from the two-dimensional sphere in ${\bf R}^{3}$ to the complex plane ${\bf C}$

A Hilbert Lemniscate Theorem in C^2

For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq 1, |Q_n(z,w)| \leq 1} approximate K and the normalized counting measures {\mu_n} associated to the finite set {P_n = Q_n = 1} converge to the pluripotential-theoretic Monge-Ampere measure for K. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable