Regularized Weighted Discrete Least Squares Approximation Using Gauss Quadrature Points

We consider polynomial approximation over the interval $[-1,1]$ by regularized weighted discrete least squares methods with $\ell_2-$ or $\ell_1-$regularization, respectively. As the set of nodes we use Gauss quadrature points (which are zeros of orthogonal polynomials). The number of Gauss quadrature points is $N+1$. For $2L\leq2N+1$, with the aid of Gauss quadrature, we obtain approximation polynomials of degree $L$ in closed form without solving linear algebra or optimization problems. In fact, these approximation polynomials can be expressed in the form of the barycentric interpolation formula when an interpolation condition is satisfied. We then study the approximation quality of the $\ell_2-$regularized approximation polynomial in terms of Lebesgue constants, and the sparsity of the $\ell_1-$regularized approximation polynomial. Finally, we give numerical examples to illustrate these theoretical results and show that a well-chosen regularization parameter can lead to good performance, with or without contaminated data.