arXiv subjects
Publications and source records attributed to Norman Levenberg.
We prove results on existence of limits in the definition of (weighted) directional Chebyshev constants at all points of the standard simplex $Σ\subset {\bf R}^d$ for (locally) regular compact sets $K\subset {\bf C}^d$.
A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ whose polynomial hull is strictly larger than $X$ but contains no analytic discs.
We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge-Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.
In their seminal paper, Berman and Boucksom exploited ideas from complex geometry to analyze asymptotics of spaces of holomorphic sections of tensor powers of certain line bundles $L$ over compact, complex manifolds as the power grows. This yielded results on weighted polynomial spaces in weighted pluripotential theory in $\Bbb{C}^d$. Here, motivated from Bayraktar's recent paper, we work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body in $(\Bbb{R}^+)^d$. These classes of polynomials need not occur as sections of tensor powers of a line bundle $L$ over a compact, complex manifold. We follow the approach in Berman and Boucksom's work to recover analogous results.
We prove a large deviation principle for the sequence of push-forwards of empirical measures in the setting of Riesz potential interactions on compact subsets K in R^d with continuous external fields. Our results are valid for base measures on K satisfying a strong Bernstein-Markov type property for Riesz potentials. Furthermore, we give sufficient conditions on K (which are satisfied if K is a smooth submanifold) so that a measure on K which satisfies a mass-density condition will also satisfy this strong Bernstein-Markov property.
We give a survey of recent results, due mainly to the authors, concerning Bernstein-Markov type inequalities and connections with potential theory.
Let $K={\bf R}^n\subset {\bf C}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,...,x_n)$ and $x^2 = x_1^2+\cdots +x_n^2$. Utilizing extremal functions for convex bodies in ${\bf R}^n\subset {\bf C}^n$ and Sadullaev's characterization of algebraicity for complex analytic subvarieties of ${\bf C}^n$ we prove the following explicit formula for the weighted extremal function $V_{K,Q}$: $$V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2})$$ where $z=(z_1,...,z_n)$ and $z^2 = z_1^2+\cdots +z_n^2$. As a corollary, we find that the Alexander capacity $T_ω({\bf R} {\bf P}^n)$ of ${\bf R} {\bf P}^n$ is $1/\sqrt 2$. We also compute the Monge-Ampère measure of $V_{K,Q}$: $$(dd^cV_{K,Q})^n = n!\frac{1}{(1+x^2)^{\frac{n+1}{2}}}dx.$$
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.
With the aid of the technique of variation of domains developed in Memoirs of Amer. Math. Soc., Vol. 209, No. 984, 2011, we characterize the pseudoconvex domains with smooth boundary in Hopf surfaces which are not Stein.
We prove a generalization of the second variation formula of the Robin function associated to a smooth variation of domains in C^N to the case of the c-Robin function associated to a smooth variation of domains in a complex manifold M equipped with a Hermitian metric and a smooth, nonnegative function c. Our purpose is that, with this added flexibility, we are able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain D in a complex homogeneous space to be Stein.
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.
We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real $n$-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.
We show that the graph $$Γ_f=\{(z,f(z))\in{\Bbb C}^2: z\in S\}$$ in ${\Bbb C}^2$ of a function $f$ on the unit circle $S$ which is either continuous and quasianalytic in the sense of Bernstein or $C^\infty$ and quasianalytic in the sense of Denjoy is pluripolar.