Maximal Convergence of Fourier Series with Respect to Bergman Orthogonal Polynomials via Their Strong Asymptotics

Let G⊂C be a bounded simply connected domain with rectifiable Jordan boundary. Denote by ϕ the conformal map of the exterior of G onto the exterior of the unit disk. For R>1, let ΓR be the level curve defined by ϕ(z)=R, and let GR denote its interior, so that G⊂GR. Suppose that f is analytic in GR. In this paper, we investigate the maximal convergence properties of the Fourier series of f with respect to the Bergman orthogonal polynomials of G. By employing the strong asymptotics of Bergman polynomials outside the domain G of orthogonality, determined by the boundary properties of G, we obtain estimates for the maximal convergence rate of the partial sums of the Fourier series of f in the uniform norm on G¯. These estimates are expressed in terms of the best polynomial approximation of f in the domain GR where f is analytic.