# Norman Levenberg's research while affiliated with Indiana University Bloomington and other places

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## Publications (37)

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubha...

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, w...

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, motivat...

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...

It is shown that there exists a Cantor set X in C^4 whose polynomially convex hull is strictly larger than X but contains no analytic discs.

We develop potential theory including a Bernstein-Walsh type estimate for functions of the form $p(g(z))q(f(z))$, where $p,q$ are polynomials and $f,g$ are holomorphic. For $g(z)=z$, such functions arise in the study of certain ensembles of probability measures and in this case we can further extend the
theory leading to probabilistic results such...

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 t...

We study the smoothness of the Siciak-Zaharjuta extremal function associated
to a convex body in $\mathbb{R}^2$. We also prove a formula relating the
complex equilibrium measure of a convex body in $\mathbb{R}^n$ to that of its
Robin indicatrix. The main tool we use are extremal ellipses.

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 \(\mathbb {C}\) with weakly admissible external fields \(Q\) and very general measures \(\nu \) on \(K\) . For this we use logarithmic potential theory in \(\mathbb {R}^{n}\) , \(n\ge...

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 dis...

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_...

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 discuss a notion of the energy of a compactly supported measure in
\mathbbCn \mathbb{C}^n for n > 1 which we show is equivalent to that defined by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion
of logarithmic energy of a measure in the complex plane
\mathbbC \mathbb{C} ; i.e., the case n = 1.
KeywordsPluripotenti...

Using the concept of Geometric Weakly Admissible Meshes together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.

We discuss three natural pseudodistances and pseudomet- rics on a bounded domain in IRN based on polynomial inequalities.

We study uniform approximation of differentiable or analytic functions of one or several variables on a compact set K by a sequence of discrete least squares polynomials. In particular, if K satisfies a Markov inequality and we use point evaluations on standard discretization grids with the number of points growing polynomially in the degree, these...

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite
diameter, and weighted transfinite diameter for sets inℂ
N
. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann.
337 (2007), 729–738) which relates the Robin function and t...

In a previous Memoirs of the AMS, vol. 92, 48, the last two authors analyzed the second variation of the Robin function-γ (t) associated to a smooth variation of domains in ℂ n for n ≥ 2. There D = ∪ t∈B(t, D(t)) ⊂ B × ℂ n was a variation of domains D(t) in ℂ n each containing a fixed point z0 and with ∂D(t) of class C ∞ for t ∈ B:= {t ∈ ℂ: |t| < ρ...

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 severa...

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 {...

We construct examples of C
∞ smooth submanifolds in ℂn and ℝn 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 co...

We show that the graph $$\Gamma_f=\{(z,f(z))\in{\Bbb C}^2: z\in S\}$$ 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.

Let K ⊂ CN be compact and let w be a nonnegative, uppersemicontinuous function on K with {z ∈ K: w(z) > 0} nonpluripolar. Let Q := - log w and define the weighted pluricomplex Green function V*K,Q(z) = lim sup ζ→z VK,Q(ζ) where VK,Q(z) := sup{u(z): u plurisubharmonic in CN, u(z) ≤ log+ |z| + C, u ≤ Q on K} (C depends on u). If w ≡ 1; i.e., Q ≡ 0, w...

Given an irreducible algebraic curves A in ℂN, let m d be the dimension of the complex vector space of all holomorphic polynomials of degree at most d restricted to A. Let K be a nonpolar compact subset of A, and for each d = 1, 2, ..., choose md points {A dj} j=1,...,md in K. Finally, let Adbe the d-th Lebesgue constant of the array {Adj}i i.e., A...

Let K be a compact convex set in C. For each point zO ε ∂K and each holomorphic polynomial p = p(z) having all of its zeros in K, we prove that there exists a point z ε K with |z -zO| ≤ 20 diam K/ √deg p such that |p′(z)| ≥ (deg p)1/2/20(diam K) |p(zO)| i.e., we have a pointwise reverse Markov inequality. In particular, ∥p′∥k ≥ (deg p)1/2/20(diam K...

LetL(D) be an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact setsK⊂RNwhose complements are John domains we prove a quantitative Runge theorem: if a functionfsatisfiesL(D)f=0 on a fixed neighborhood ofK, we estimate the sup-norm distance fromfto the polynomial solutions of degree at...

We study solutions of the equation L(D) = 0, where L(D) is an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact sets K ⊂ RN with connected complement we prove a Bernstein theorem: if a function ƒ on K can be extended to a solution of the equation on an open neighborhood of K, then the...

The classical Bernstein theorem states that a continuous function f on [-1,1] extends to a holomorphic function on an open neighborhood of [-1,1] in ℂ if and only if lim sup n→∞ d n 1/n <1, where d n ≡inf∥f- p n ∥ [-1,1] : p n polynomial of degree ≤ n and ∥·∥ denotes the supremum norm. We outline four proofs of this theorem, and indicate how these...

We use geometric methods to calculate a formula for the complex Monge-Ampere measure (ddcVK)n, for K ⋐ Rn ⊂ Cn a convex body and VK 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 estim...

## Citations

... , |z d |). As in [3,4,7], we make the assumption on P that ⊂ k P for some k ∈ Z + . (1.2) In particular, 0 ∈ P. ...

... There are now many known counterexamples due to several mathematicians. By combining our results on pairs of polynomially convex sets whose union has nontrivial hull with a construction of nontrivial hulls without analytic discs in the paper of the author and Norman Levenberg [9], we will establish results on pairs of polynomially convex sets whose union has nontrivial hull without analytics discs (Theorems 6.1 and 6.2 and Corollary 6.3). ...

... The spaces Poly(nC) and C-extremal function V C,K (z) := sup 1 deg C p log + |p(z)|: p K ≤ 1 were defined in [1] and further studied in [7]. * The equilibrium measure (dd c V C,K ) N and C-transfinite diameter δ C (K) were studied in [2], using the L 2 energy methods of Berman-Boucksom pioneered in [3,4]. The Berman-Boucksom methods work more naturally in a general weighted setting involving an admissible weight function w : K → [0, ∞) and its logarithm Q = − log |w|. ...

... In the case of the 2d Coulomb gas, this will be illustrated using the well-known notion of Bernstein-Markov inequalities. This notion can be extended to a general pair interaction potential W as follows (the case of the Riesz gas was introduced in [18]). Definition 4.5. ...

... Hence, the special case of (1.6) with f (x) = x θ is also called the Muttalib-Borodin ensemble, see [19,20,30,37,40] for example. Biorthogonal ensembles and variations thereof are also considered in [5,8,9,14,17,22,24,31,32,39]. ...

... The Bernstein-Markov property is a classical concept and was studied thoroughly in [1,2,4],. . . The reader is referred to [3] for an authoritative survey on Bernstein-Markov property and its numerous applications to approximation theory. One use of this property is the possibility to approximate the (global) extremal function of K by functions of the form 1 deg p log |p| where p are polynomials that form an orthonormal system for L 2 (μ) (see [4]). ...

... It is worth saying that explicit formulas for V * E and μ * E have been computed only for few instances E, see, e.g., [1][2][3]19,24,41,48]. Numerical approximation schemes were introduced in [49]. ...

... Further Turán-type inequalities for polynomials with restricted zeros may be found in Erőd [4], Bojanov [1], Levenberg and Poletsky [13], Erdélyi [2], Révész [14], [15], Glazyrina and Révész [8], Govil and Mohapatra [9]. Weighted Turán-type inequalities were considered by Xiao and Zhou [21], Wang and Zhou [20]. ...

... The proof relies on an extension result with growth control of psh functions on analytic subvarieties of Stein manifolds [CGZ13,Theorem A]. This in turn used methods of Coltoiu based on Runge domains [Co91,Proposition 2], and of Sadullaev [Sa82] (see also [BL03,Theorem 3.2]). ...

... Here for K ⊂ C d compact, (1.1) where p is a nonconstant holomorphic polynomial; and for a continuous complex-valued function f on K, D n (f, K) := inf{||f − p n || K : p n ∈ P n } where P n is the space of holomorphic polynomials of degree at most n. See [1] for a survey and history of Theorem 1.1 in both one and several complex variables. ...