Regularized Weighted Discrete Least Squares Approximation by Orthogonal Polynomials

We consider polynomial approximation over the interval $[-1,1]$ by a class of regularized weighted discrete least squares methods with $\ell_2-$regularization and $\ell_1-$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 $L\leq2N+1$. With the aid of Gauss quadrature, we obtain approximation polynomials of degree $L$ in closed form without solving linear algebra or optimization problem. As a matter of fact, these approximation polynomials can be expressed in the form of barycentric interpolation when the interpolation condition is satisfied. We then study the approximation quality of $\ell_2-$regularization approximation polynomial, especially on the Lebesgue constant. Moreover, the sparsity of $\ell_1-$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.