arXiv subjects
Publications and source records attributed to N. Levenberg.
Using recent results of Berman and Boucksom we show that for a non-pluripolar compact set K in C^d and an admissible weight function w=e^{-ϕ} any sequence of so-called optimal measures converges weak-* to the equilibrium measure μ_{K,ϕ} of (weighted) Pluripotential Theory for K,ϕ.
We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in C^N. An ingredient is a formula of Rumely which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in C^N. Our results extend to certain weights and measures defined on cones in R^N.
We use geometric methods to calculate a formula for the complex Monge-Ampère measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies $K$. We apply this to show that two methods for deriving Bernstein-Markov-type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area ellipses determining the complex Monge-Ampère solution $V_K$.
We discuss the Siciak-Zaharjuta extremal function of a real convex body in C^n, a solution of the homogeneous complex Monge-Ampere equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadric curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain.
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 {μ_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.
We show that on the curves y=e^{t(x)} where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four. Specifically, for the real interval [a,b] there is a constant C such that max_{x\in [a,b]}|\frac{d}{dx}P(x,e^{t(x)})|\leq C(deg(P))^{4} max_{x\in [a,b]}|P(x,e^{t(x)})| for all bivariate polynomials P(x,y).