On the norm of the hyperinterpolation operator on the unit disk

In this article we study the properties of the hyperinterpolation operator on the unit disk D in R, approximating the orthogonal projection of a function onto the family of polynomials of degree n. A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results then prove the uniform convergence of the hyperinterpolation approximation of functions in the class C (D), > 0. We show how hyperinterpolation can be used in connection with the Kumar-Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disk (discrete Galerkin method). Finally we present numerical examples which show that the discrete Galerkin method converges faster than O n k , for every k 2 N, if the solution of the nonlinear Poisson equation in is C1 (D). Keywords: Hyperinterpolation operator, discrete Galerkin method, projection norm, nonlinear Poisson equation AMS subject classi…cation: 65R20, 65N35, 35J60, 41A55.