Treffer: On the normal matrix of the polynomial LS problem over the Chebyshev points

Let \(V = [x_i^{j-1}]\) be an \(n\times m\) (\(n\geq m\)) Vandermonde matrix with the Chebyshev nodes \(x_i = \cos(\pi (2i-1) /2n)\). This paper provides an explicit formula for the Cholesky factor of the matrix \(V^T V\), leading to an \(O(m (m+n) )\) method for solving polynomial approximation problems. It turns out that the entries of the Cholesky factor and its inverse can be represented as integers divided by powers of two. Numerical experiments suggest that this approach is more efficient but also sometimes less accurate than methods based on orthogonal polynomials for solving approximation problems.