The Christoffel-Darboux kernel

Abstract

Abstract. A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.

1D log gases and the renormalized energy: crystallization at vanishing temperature

“…Theorem 6. • the proof that the minimum of W is achieved by the one-dimensional regular lattice Z, called the "clock distribution" in the context of orthogonal polynomial ensembles [42]. This is in contrast with the dimension 2 where the identification of minimizers of W is still open (but conjectured to be "Abrikosov" triangular lattices).…”

Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications

“…Let μ be a finite positive Borel measure supported on an infinite subset of R. It is well-known that the polynomial kernels (also called reproducing, Christoffel-Darboux or Dirichlet kernels) associated with the sequences of orthogonal polynomials corresponding to μ are frequently used as a basic tool in spectral analysis, convergence of orthogonal expansions [2,23,27], and other aspects of mathematical analysis (see [26] and the references therein). In the setting of orthogonal polynomial theory these kernels have been especially used by Freud and Nevai [4,21,22] and, more recently, the remarkable Lubinsky's works [9,10] have caused heightened interest in this topic.…”

Point mass insertion on the real line and non‐exponential perturbation of the recursion coefficients

“…Moreover, when considering the effect of varying the weight of discrete point masses on orthogonal polynomials (both on R and ∂D), Simon proved that it will result in exponential perturbation of the recursion coefficients (see Corollary 24.4 and Corollary 24.3 of [15]). …”