A real variable calculus for planar orthogonal polynomials

This note is part of a project in which we aim to obtain a global calculus for polynomial Bergman kernels in the plane. We aim to find a characterization of the diagonal restriction of the kernel as the unique solution to a certain non-linear soft Riemann-Hilbert problem. It is well known that the diagonal restriction of the polynomial Bergman kernel equals a sum of the squared moduli of the normalized orthogonal polynomials. Here, we characterize these squared moduli in terms of a real variable non-linear soft Riemann-Hilbert problem for the Laplacian (in place of the $\bar\partial$-operator), and find a direct algorithm to produce suitable approximants. The appearance of squared moduli creates a non-linear effect which was not present in earlier work.Comment: 24 page