Weighted Least Squares Approximation by Orthogonal Polynomials with a Regularization Term

In this paper we consider weighted least squares approximation by orthogonal polynomials with a regularization term on $[-1,1]$. The models are better than classical least squares even most regularized least squares due to their lead to closed-form solutions. The closed-form solution for models with $\ell_2$-regularization term derive the closed-form expression of the Lebesgue constant for such an approximation. Moreover, we give bounds for the number of nonzero elements of the solution for models with $\ell_1$-regularization term, which shows the sparsity of the solution.