Quadrature at fake nodes

We investigate the use of the so-called mapped bases or fake nodes approach in the framework of numerical integration. We show that such approach is able to mitigate the Gibbs phenomenon when integrating functions with steep gradients. Moreover, focusing on the optimal properties of the Chebyshev-Lobatto nodes, we are able to analytically compute the quadrature weights of the fake Chebyshev-Lobatto nodes. Such weights, quite surprisingly, coincide with the composite trapezoidal rule. Numerical experiments show the effectiveness of the proposed method especially for mitigating the Gibbs phenomenon without the need of resampling the given function.