[Paper Review] On the Convergence of Optimal Measures

Thomas Bloom, Len Bos+2 more|ArXiv.org|Aug 6, 2008

Geometry and complex manifolds7 references4 citations

TL;DR

This paper establishes the weak-* convergence of optimal measures—defined as probability measures minimizing the maximal Christoffel function on a non-pluripolar compact set $K \subset \mathbb{C}^d$ with an admissible weight $w = e^{-\phi}$—to the equilibrium measure $\mu_{K,\phi}$ in weighted pluripotential theory. Using Berman and Boucksom's results on energy functionals and the Rumely formula, it proves that the sequence of optimal measures converges to the equilibrium measure as the degree $n \to \infty$, extending known convergence for Fekete points and Leja sequences.

ABSTRACT

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.

Motivation & Objective

• To establish the weak-* convergence of optimal measures to the equilibrium measure in weighted pluripotential theory for non-pluripolar compact sets in $\mathbb{C}^d$.
• To extend known convergence results for Fekete points and Leja sequences to a broader class of optimal measures defined via extremal minimization of the Christoffel function.
• To provide a unified framework linking optimal measures, determinant maximization of Gram matrices, and equilibrium measures through pluripotential-theoretic tools.

Proposed method

• Define optimal measures of degree $n$ as probability measures on $K$ that minimize $\max_{z \in K} \sqrt{K_n^\mu(z)}$, where $K_n^\mu(z)$ is the diagonal of the reproducing kernel for holomorphic polynomials of degree $\leq n$.
• Use the equivalence between minimizing the maximal Christoffel function and maximizing the determinant of the Gram matrix for any fixed basis of $\mathcal{P}_n$, via the factorization $G_n^\mu = V_n^\mu (V_n^\mu)^*$.
• Apply the Rumely formula for the transfinite diameter $\delta^w(K)$ in terms of mixed energy $\mathcal{E}(V_{K,\phi}^*, V_T)$, and use Berman and Boucksom's derivative formula for the energy with respect to weight perturbations.
• Introduce a one-parameter family of weights $w_t = w \exp(-t u)$ and analyze the associated function $f_n(t) = -\frac{1}{2m_n} \log \det G_n^{\mu_n, w_t}$, showing $\liminf_{n \to \infty} f_n(t) \geq -\log \delta^{w_t}(K)$.
• Leverage Lemma 3.2 from Berman and Boucksom [BB2] to conclude $\lim_{n \to \infty} f_n'(0) = g'(0)$, where $g(t) = -\log \delta^{w_t}(K)$, which yields convergence of the measures via integration against test functions.
• Conclude that $\mu_n \to \mu_{K,\phi}$ weak-* by matching the weak-* limit of $\mu_n$ to the measure $\mu_{K,\phi}$ via the identity $\int_K u \, d\mu_n \to \int_K u \, d\mu_{K,\phi}$ for all continuous $u$.

Experimental results

Research questions

• RQ1 Does the sequence of optimal measures for a non-pluripolar compact set $K \subset \mathbb{C}^d$ with an admissible weight $w = e^{-\phi}$ converge weak-* to the equilibrium measure $\mu_{K,\phi}$ as $n \to \infty$?
• RQ2 Can the convergence of optimal measures be established using the derivative of the energy functional in pluripotential theory, particularly via the Berman-Boucksom derivative formula?
• RQ3 How does the minimization of the maximal Christoffel function relate to the maximization of the determinant of the Gram matrix for polynomial bases in weighted $L^2$-spaces?
• RQ4 Is the weak-* convergence of optimal measures a generalization of known convergence results for Fekete points and Leja sequences in polynomial interpolation?

Key findings

• The sequence of optimal measures $\mu_n$ for $K$ and weight $w = e^{-\phi}$ converges weak-* to the equilibrium measure $\mu_{K,\phi}$ as $n \to \infty$, establishing a fundamental convergence result in weighted pluripotential theory.
• Optimal measures are characterized by the property $\max_{z \in K} K_n^\mu(z) = N = \binom{d+n}{n}$, which is the minimal possible value for the maximal Christoffel function.
• The optimal measure also maximizes the determinant of the Gram matrix $G_n^\mu(B_n)$ over all probability measures on $K$, for any fixed basis $B_n$ of $\mathcal{P}_n$, linking extremal problems in approximation theory and orthogonal polynomials.
• The convergence is proven via the derivative of the transfinite diameter $\delta^{w_t}(K)$ with respect to weight perturbations, using the Rumely formula and the Berman-Boucksom derivative identity.
• The weak-* convergence follows from the convergence of the derivative of the logarithmic determinant functional $f_n(t)$ at $t=0$, which matches the derivative of the energy functional $g(t)$, implying measure convergence via Lemma 3.2 from [BB2].
• The result generalizes known convergence for Fekete points and Leja sequences to the broader class of optimal measures, showing that such measures are asymptotically equidistributed with respect to the equilibrium measure $\mu_{K,\phi}$.