Christoffel–Darboux kernels, mollification and density recovery
With Mauricio Velasco, Didier Henrion
The Christoffel–Darboux (CD) function, obtained from the CD kernel, is a central tool in approximation theory, orthogonal polynomials and moment-based methods. It efficiently encodes geometric and statistical information about a measure, with applications to support inference, anomaly detection, density approximation, optimal sampling and machine learning.
Together with Mauricio Velasco and Didier Henrion, we introduced mollified CD kernels on algebraic varieties. They improve support recovery and yield convergence rates for density recovery without requiring knowledge of the equilibrium measure of the support, which the classical method needs.
The Fekete problem
With Diego Armentano, Federico Carrasco, Marcelo Fiori, Pedro Raigorodsky, Matías Valdés, Mauricio Velasco
The Fekete problem asks, for a given natural number N, for N points on the sphere that maximize the product of their pairwise distances — equivalently, that maximize the logarithmic energy of the configuration. Smale's 7th problem asks for configurations whose logarithmic energy is within order log(N) of the optimum; exact solutions are known only for N = 2, 3, 4, 5, 6 and 12.
With Diego Armentano, Federico Carrasco, Marcelo Fiori, Pedro Raigorodsky, Matías Valdés and Mauricio Velasco, we study the critical configurations as an algebraic variety, using computational tools such as Gröbner bases to count and classify critical points for small N.
