The purpose of this short note is to show that the Christoffel–Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving dual problem semi-algebraic D-optimal experimental design statistics. It uses only elementary notions convex analysis. Geometric interpretations algorithmic consequences are mentioned.

We consider the problem of reconstructing an unknown function $$u\in L^2(D,\mu )$$ from its evaluations at given sampling points $$x^1,\dots ,x^m\in D$$ , where $$D\subset {\mathbb {R}}^d$$ is a general domain and $$\mu $$ probability measure. The approximation picked linear space $$V_n$$ interest $$n=\dim (V_n)$$ . Recent results (Cohen Migliorati in SMAI J Comput Math 3:181–203, 2017, Doostan...

A reduction of Benney's equations is constructed corresponding to Schwartz-Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian σ-function of the curve.

We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x, x0)Kn(x0, x0)−1 dρ(x) → δx0 , where Kn is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.

We show that the computational complexity of Riemann mappings can be bounded by the complexity needed to compute conformal mappings locally at boundary points. As a consequence we get first formally proven upper bounds for Schwarz-Christoffel mappings and, more generally, Riemann mappings of domains with piecewise analytic

We study the stability of convergence of the ChristoffelDarboux kernel, associated with a compactly supported measure, to the sine kernel, under perturbations of the Jacobi coefficients of the measure. We prove stability under variations of the boundary conditions and stability in a weak sense under ` and random ` diagonal perturbations. We also show that convergence to the sine kernel at x imp...

The classical Brunn-Minkowski theory for convex bodies was developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As a special case of mixed volumes, the Quermassintegrals are important geometrical quantities of a convex body, and surface area measures are local versions of Quermassintegrals. The Christoffel-Minkowski problem concerns with the existen...