We consider polynomial approximation over the interval [-1,1] by a class of regularized weighted discrete least squares methods with l2-regularization and l1-regularization terms, respectively. It is merited to choose classical orthogonal polynomials as basis sets of polynomial space with degree at most L. As node sets we use zeros of orthogonal polynomials such as Chebyshev points of the first kind, Legendre points. The number of nodes, say N+1, is chosen to ensure L<=2N+1. With the aid of Gauss quadrature, we obtain approximation polynomials of degree L in closed form without solving linear algebra or optimization problem. It can be shown that they are extentions of Wang-Xiang formula for polynomial interpolation. We then study theapproximation quality of l2-regularization approximation polynomial and the sparsity of l1-regularization approximation polynomial, respectively. Finally, we give numerical examples to illustrate these theoretical results and show that well-chosen regularization parameter can provide good performance approximation, with or without contaminated data.
