On Kosloff Tal-Ezer least-squares quadrature formulas

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on arbitrary grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes with a low-order Lebesgue constant. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and Least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter $\alpha$ ensures a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for $\alpha$ converging to $0^{+}$ and $1^{-}$, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that some static choices of the map's parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way.