Abstract The Christoffel–Darboux kernels for orthogonal polynomials in several real variables are investigated within the context of three-term relation reformulated this purpose. As suggestive examples orthogonality, we propose to discuss two simple algebraic cases: unit circle and Bernoulli lemniscate.

We show that the empirical Christoffel function associated with a cloud of finitely many points sampled from distribution, can provide simple tool for supervised classification in data analysis, good generalization properties.

4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.2. 4.3. Gaussian quadrature with preassigned nodes Christoffel's work and related developments Kronrod's extension of quadrature rules Gaussian quadrature with multiple nodes The quadrature formula of Turan Arbitrary multiplicities and preassigned nodes Power-orthogonal polynomials Constructive aspects and applications Further miscellaneous extensions Product-type...

In this note we present some results on the Calkin-Wilf tree of irreducible fractions, giving an insight on the duality relating it to the Stern-Brocot tree, and proving noncommutative versions of known results relating labels of the CalkinWilf trees to hyperbinary expansions of positive integers. The main tool is the Christoffel tree introduced in a paper by Berstel and de Luca.