  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.
