Estimates of Christoffel function on multivariate domains
We consider Christoffel functions defined on some compact multivariate domain with non-empty interior. Christoffel functions have applications in different problems of approximation theory, harmonic analysis, numerical analysis, statistical physics, random matrix theory and spectral theory and other areas of mathematics. Understanding the structure of orthonormal basis of polynomials on a general multivariate domain, which can be used to compute Christoffel function, can be a very challenging task. Known results are mostly limited only to very specific domains. In this thesis, we obtain some new estimates for Christoffel functions on multivariate domains from several general classes. In most situations we managed to compute the Christoffel function up to a constant factor avoiding explicit computation of the corresponding orthonormal system of polynomials. One of the key tools we apply is the technique of comparison between different domains. Our first result is a lower bound on Christoffel function on planar convex domains in terms of an adaptation of the parallel section function of the domain. For a certain class of planar convex domains this allows us to compute the pointwise behavior of Christoffel function using previously known upper bound. Our second result is a computation up to a constant factor of the Christoffel function on planar domains with boundary consisting of finitely many 𝐶2 curves such that each corner point of the boundary has interior angle strictly between 0 and π. Finally, we estimate up to the constant the Christoffel function on any simple polytope in multivariate space.
Christoffel Function, Multivariate Domains